Restor. I-131 Upt. + Selpercatinib in RET F-P RAI-R TC

Inclusion Criteria: * Participants must have histologically- or cytologically-confirmed follicular-derived nonanaplastic thyroid cancer that is metastatic and/or unresectable AND harbors a known oncogenic RET gene fusion, performed in a Clinical Laboratory Improvement Amendments (CLIA)-certified or equivalently accredited diagnostic laboratory. * Tumor tissue or liquid biopsy-based next-generation sequencing (NGS), quantitative polymerase chain reaction (qPCR), and fluorescence in situ hybridization (FISH) for RET gene fusion detection will be permitted. * Participants ≥ 18 years of age must have measurable disease, per Response Evaluation Criteria in Solid Tumors (RECIST) v1.1. \-- Participants 12 to \<18 years of age may enroll with either evaluable (i.e. anatomically visible tumor on cross sectional imaging, but tumors do not need to be \>1 cm) or measurable disease per RECIST v1.1. * Participants should have no single tumor deposit exceeding 4.0 cm in the greatest dimension. * Participants must have RAI-refractory disease, as defined by: * The malignant tissue does not concentrate RAI on a whole body scan following radioiodine administration * The tumor tissue has lost the ability to concentrate RAI on a whole body scan following radioiodine administration after previous evidence of uptake at any earlier timepoint, * RAI is taken up in some but not all tumor deposits on a whole body scan following radioiodine administration and/or * There is progressive disease per RECIST v1.1 despite RAI uptake on a whole body scan following radioiodine administration and/or * For patients 12 to \<18 years of age, there is persistent anatomically visible tumor on cross sectional imaging following prior therapeutic radioiodine administration * Participants may have received no more than a total cumulative RAI dose for treatment (not including RAI given for diagnostic purposes only) of 500 mCi (18.5 GBq). * Participants must have asymptomatic or minimally symptomatic disease, as judged by the treating investigator. For example, patients with bone metastasis associated with mild pain not requiring narcotics for pain control, or patients with lung metastasis associated with mild cough that does not limit the participant's activities, may be considered minimally symptomatic. If confirmation of this criterion is needed, discussion with the protocol chairperson is required prior to enrollment. * Prior external beam radiotherapy is allowed. For participants with disease limited to a prior radiotherapy field, this must be considered measurable per RECIST v1.1. * Participants may have had no more than one prior systemic therapy for RAI-refractory thyroid cancer, including lenvatinib, sorafenib, or other MKIs. This also includes chemotherapy and/or targeted therapy administered within a clinical trial, but does not include I-131 NaI or levothyroxine. Prior RET-specific kinase inhibitor therapy, such as selpercatinib and pralsetinib, is not allowed. * At least 28 days must have passed since any prior radiation or major surgery. At least 28 days must have passed since any prior systemic therapy for thyroid cancer, excluding thyroid hormone replacement. * Participants must be ≥18 years of age. Patients as young as 12 years of age will be allowed if permitted by local regulatory authorities and institutional review boards. \-- All patients of 12 to \< 18 years of age will be asked to provide assent and the legally designated representative will be asked to provide written consent. * Participants ≥ 16 years of age must have an Eastern Cooperative Oncology Group (ECOG) performance status ≤ 1. Participants 12 to \<16 years of age must have a Lansky Performance Score of ≥ 60. * Participants must have a life expectancy greater than 12 months. * Participants must have the ability to swallow medications and have no gastrointestinal abnormality that may alter medication absorption. * Participants must have adequate organ and marrow function as defined below. Bloodwork must be collected within 7 days prior to Cycle 1 Day 1: * leukocytes ≥3,000/mcL * absolute neutrophil count (ANC) ≥1,500/mcL * platelets ≥100,000/mcL * hemoglobin ≥9 g/dL * total bilirubin ≤ 1.5 × institutional upper limit of normal (ULN) OR \<3.0 x institutional ULN for participants with Gilbert syndrome * AST(SGOT)/ALT(SGPT) ≤2.5 × institutional ULN OR ≤5 × institutional ULN if the liver has tumor involvement * creatinine ≤ institutional ULN OR creatinine clearance ≥30 mL/min * Participants must have serum potassium, calcium and magnesium levels within institutional range of normal (may be receiving supplements). Bloodwork must be collected within 7 days prior to Cycle 1 Day 1. * Known human immunodeficiency virus (HIV)-infected participants on effective anti- retroviral therapy with an undetectable viral load within 6 months are eligible for this trial. * For participants with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. * Participants with a history of hepatitis C virus (HCV) infection must have been treated and cured. For participants with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. * Participants with treated brain metastases are eligible if the patient is asymptomatic AND follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression. * Participants with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. For example, participants with nonmelanoma skin cancer, carcinoma in situ of the cervix or malignancy diagnosed ≥2 years previously and not currently receiving anticancer treatment are eligible. Patients receiving adjuvant hormone therapy for breast or prostate cancer with no evidence of disease are eligible. * Participants with known history or current symptoms of cardiac disease or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, participants should be class 2B or better. * Selpercatinib may impair fertility in men and women. The effects of selpercatinib on the developing human fetus are unknown. For this reason and because selpercatinib as well as other therapeutic agents used in this trial may be teratogenic, women of child-bearing potential (WOCBP) and men must agree to use highly effective contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation and for 6 months after the last dose of study drug. Unless not allowed by local regulations, WOCBP who are abstinent (if this is complete abstinence, as their preferred and usual lifestyle) or in a same-sex relationship (as part of their preferred and usual lifestyle) must agree to either remain abstinent or stay in a same-sex relationship without sexual relations with males. Periodic abstinence (e.g. calendar, ovulation, symptothermal, postovulation methods), declaration of abstinence just for the duration of the trial, and withdrawal are not acceptable methods of contraception. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men with partners who are WOCBP must agree to use a highly effective contraceptive method during treatment with the study drugs and for 6 months following the last dose of study drug. * WOCBP must have a negative pregnancy test (serum or urine) within 7 days prior to initiating study treatment, and not be breast-feeding within two months prior to study entry, during treatment, and for ≥1 week after the last dose of study therapy. * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Participants who have had chemotherapy, MKI or radiotherapy within 4 weeks. * Participants who have had I-131 NaI treatment within 12 months. * Participants who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> Grade 1) with the exception of alopecia. * Participants who are receiving any other investigational agents. * Participants with symptomatic CNS metastasis or lesions threatening for spinal cord compression are not eligible. * Participants with clinically significant active cardiovascular disease or history of Torsades de pointes, history of myocardial infarction within 6 months prior to planned start of study treatment, or prolongation of the corrected QT interval by Fridericia's formula (QTcF) \>470 msec on more than one ECG during Screening. Correction of suspected drug-induced QTcF prolongation may be attempted at the investigator's discretion if clinically safe to do so. * Participants with uncontrolled hypertension at screening, as defined by \>160/95 mm Hg. * Participants with uncontrolled hypertension at screening may be re-screened after appropriate medical therapy for hypertension. * Participants with an active uncontrolled systemic bacterial, viral, or fungal infection or serious ongoing intercurrent illness, such as hypertension or diabetes, despite optimal treatment, a clinical diagnosis or symptoms of interstitial lung disease, or other serious medical conditions which in the medical judgment of the investigator would prevent the patient from safely participating (screening for chronic conditions is not required). * Participants with uncontrolled symptomatic hyperthyroidism or hypothyroidism. * Participants with symptomatic hypercalcemia or hypocalcemia. * Participants with active hemorrhage or at significant risk for hemorrhage. * Participants who are taking a concomitant medication that is known to cause QTc prolongation ). * Participants with clinically significant active malabsorption syndrome or other condition likely to affect gastrointestinal absorption of the drug. * Participants with other uncontrolled serous intercurrent illness that would interfere with the ability to proceed with study therapy.