Enhancing Radioiodine Incorporation Into Radio Iodine Refractory Thyroid Cancers With MAPK Inhibition

Inclusion Criteria: * Patients must have histologically or cytologically confirmed radioiodine refractory metastatic thyroid carcinoma of follicular origin (including papillary and its respective variants). * Confirmation of the mutation status of BRAF gene (primary tumor, recurrent tumor, or metastasis) . * Patients who do not undergo a systemic treatment with sorafenib or lenvatinib or chemotherapy or with other TKIs or other investigational drugs. * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥ 15 mm with CT scan, MRI, or calipers by clinical exam. Tumors in previously irradiated fields may be considered measureable if there is evidence of tumor progression after radiation treatment. * RAI-refractory disease on structural imaging, defined as following: A metastatic lesion that is not radioiodine-avid on a diagnostic or therapeutic radioiodine scan performed less than 1 year prior to enrollment in the current study, There are no size limitations for the index lesion used to satisfy this entry criterion. * No recent treatment for thyroid cancer as defined as: 1. No prior 131I therapy is allowed \< 6 months prior to initiation of therapy on this protocol. A diagnostic study using \< 400 MBq (±100) of 131I is not considered 131I therapy. 2. No external beam radiation therapy \< 4 weeks prior to initiation of therapy on this protocol. (Previous treatment with radiation for any indication is allowed if the investigator judges that the previous radiation does not significantly compromise patient safety on this protocol.) 3. No chemotherapy or targeted therapy (e.g., tyrosine kinase inhibitor) is allowed \< 4 weeks prior to the initiation of therapy on this protocol. * Age ≥ 18 years * ECOG performance status ≤ 2. * Life expectancy of greater than 3 months. Able to swallow and retain orally-administered medication and does not have any clinically significant gastrointestinal abnormalities that may alter absorption such as malabsorption syndrome or major resection of the stomach or bowels. * Patients must have normal organ and bone marrow function as defined below: * Absolute neutrophil count (ANC) \> 1.5x10\^9/L * Hemoglobin ≥ 9 g/dL * Platelets ≥ 100 x 10\^9/L * Albumin ≥ 2.5 g/dL * Total bilirubin ≤ 1.5x institutional ULN * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 2x institutional ULN unless it is related to the primary disease * creatinine ≤ 1.5 mg/dL OR calculated creatinine clearance (Cockcroft-Gault formula) ≥ 50 mL/min OR 24-hour urine creatinine clearance ≥ 50 mL/min * Negative pregnancy test within 7 days prior to starting the study premenopausal women. Women of non-childbearing potential may be included without pregnancy test if they are either surgically sterile or have been postmenopausal for ≥ 1 year. * Fertile men and women must use an effective method of contraception during treatment and for at least 6 months after completion of treatment as directed by their physician. Effective methods of contraception are defined as those, which result in a low failure rate (i.e., less than 1% per year) when used consistently and correctly (for example implants, injectables, combined oral contraception or intra-uterine devices). At the discretion of the investigator, acceptable methods of contraception may include total abstinence in cases where the lifestyle of the patient ensures compliance. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, postovulation methods\] and withdrawal are not acceptable methods of contraception.) * Ability to understand and the willingness to sign a written informed consent document. * Patients must agree to undergo to research biopsy of a malignant lesion if the mutation status cannot be proven through archival tissue specimen. * Availability of archival tumor tissue from the thyroid cancer primary or metastasis (a tissue block or a minimum of 30 unstained slides would be required. Patients with less archival tissue available may still be eligible for the study after discussion with the Principal Investigator). This does not apply to patients who undergo a biopsy. Exclusion Criteria: * Concomitant malignancies or previous malignancies of life-limiting potential within the last 3 years. * Use of other investigational drugs within 28 days preceding the first dose of drug treatment during this study. * Known leptomeningeal or brain metastases or spinal metastases. * Have a known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to trametinib and/or to dabrafenib or other known contents of the two investigational drugs. * History or evidence of cardiovascular risk including any of the following: * History or evidence of current, clinically significant uncontrolled arrhythmias (exception: patients with controlled atrial fibrillation for \> 30 days prior to the initiation of therapy on this protocol are eligible). * History of acute coronary syndromes (specifically, myocardial infarction and unstable angina), severe/unstable angina, coronary angioplasty, or stenting within 6 months prior to the initiation of therapy on this protocol. * History of symptomatic congestive heart failure within 6 months prior to the initiation of therapy on this protocol. * History of cerebrovascular attack or transient ischemic attack within 6 months prior to the initiation of therapy on this protocol. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection or psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant, lactating, or breast feeding women. * Patients unable to follow a low iodine diet or requiring medication with high content in iodide (amiodarone). * Patients who received iodinated intravenous contrast as part of a radiographic procedure within 3 months of study registration. Those that have had iodinated intravenous contrast within this time frame may still be eligible if a urinary iodine analysis reveals that the excess iodine has been adequately cleared after the last intravenous contrast administration. * Unwillingness or inability to comply with study and follow-up procedures. * Disorders of eye background as listed in the respective study drug prescribing information. * Patients with pancreatitis, prolonged QTc-time on EKG, uncontrolled hypertension, thrombosis or high risk of bleedings. * Condition of patient which is critical to participate in this study in the discretion of the PI.