Lenvatinib in Locally Advanced Invasive Thyroid Cancer

Inclusion Criteria: * 18 years of age at the time of informed consent and willing and able to provide written informed consent for the trial. * Adult participants who are either initially diagnosed with locally advanced thyroid neoplasm or have experienced persistent or recurrent thyroid and/or cervical nodal recurrent DTC (participants with M1 disease are allowed, AJCC 8th edition stage I-IVb)), including: * a. Papillary thyroid carcinoma (PTC) - classical and variants * Follicular variant * Variants including but not limited to tall cell, columnar cell, cribriform-morular, solid, oxyphil, Warthin's-like, trabecular, tumor with nodular fasciitis-like stroma, Hürthle cell variant of papillary carcinoma * b. Follicular thyroid carcinoma (FTC) * c. Hürthle cell carcinoma * d. Poorly differentiated thyroid carcinoma * e. Cytologically confirmed thyroid neoplasm, Bethesda 3, 4 and 5 * Evidence of extrathyroidal extension and/or locally invasive disease and deemed at risk for R2 resection by treating team on clinical and/or fiberoptic examination and/or radiographic evaluation in the primary or recurrent setting. Evidence of "at risk for R2 resection" includes: * a. Vocal cord paralysis by fiberoptic examination * b. Extrathyroid and/or extranodal extension on CT or MRI, including tracheal and/or laryngeal cartilage invasion, esophageal involvement, and/or involvement of perithyroid muscles (e.g. strap, sternocleidomastoid, inferior constrictor muscles) or bone involvement * c. Extension into the mediastinum with visceral and/or vascular involvement * d. Involvement of the carotid artery or other major vessel by 180 degrees or more (exclusive of complete encasement) * e. Other factors that make the participant to be "at risk for R2 resection" may be allowed, after discussion with the study's principal investigator. * Measurable disease by RECIST v1.1. * Eastern Cooperative Oncology Group (ECOG) performance status (PS) ≤ 1 and no medical contraindication to surgery. * Adequately controlled blood pressure. * Blood pressure ≤150/90 with or without antihypertensive medications at screening * Adequate end-organ function (including bone marrow, coagulation, renal, liver and cardiac) 28 days prior to the study registration as defined below: * leukocytes ≥3,000/mcL * absolute neutrophil count ≥1,500/mcL * platelets ≥100,000/mcL * total bilirubin ≤1.5 x institutional upper limit of normal, unless attributed to Gilberts syndrome * AST/ALT/Alk Phos ≤3 x institutional upper limit of normal * INR ≤1.5 x institutional upper limit of normal * creatinine within normal institutional limits OR * creatinine clearance ≥30 mL/min per Cockcroft-Gault formulation * The cycle 1 day 1 labs need to re-meet eligibility criteria for treatment * Ability to swallow pills. * Females must not be lactating or pregnant at baseline (as documented by a negative betahuman chorionic gonadotropin \[ß-hCG\] test with a minimum sensitivity of 25 IU/L or equivalent units of ß-hCG. A separate baseline assessment is required if a negative screening pregnancy test was obtained more than 72 hours before the first dose of study drug. * Note: All females will be considered to be of childbearing potential unless they are postmenopausal (amenorrheic for at least 12 consecutive months, in the appropriate age group and without other known or suspected cause) or have been sterilized surgically (ie, bilateral tubal ligation, total hysterectomy or bilateral oophorectomy, all with surgery at least 1 month before dosing). * Females of childbearing potential must not have had unprotected sexual intercourse within 30 days before study entry and must agree to use a highly effective method of contraception (eg, total abstinence, an intrauterine device, a contraceptive implant, an oral contraceptive, or have a vasectomized partner with confirmed azoospermia) throughout the entire study period and for 30 days after study drug discontinuation. Females who are using hormonal contraceptives must have been on a stable dose of the same hormonal contraceptive product for at least 4 weeks before dosing and must continue to use the same contraceptive during the study and for 30 days after. Exclusion Criteria: * Diagnosis of medullary thyroid carcinoma or anaplastic (undifferentiated) thyroid carcinoma. * Radiographically identified following findings: * intraluminal airway tumor * complete carotid encasement/infiltration * Active hemoptysis (bright red blood ≥ 1/2 teaspoon) or other uncontrolled bleeding within 21 days prior to the study registration. * Arterial/venous thromboembolic events in the last 12 months Treatment within 30 days prior to study registration with anticoagulant or antiplatelet therapy, apart from aspirin 81 mg daily. * Prior radiotherapy to the neck. * Prior treatment with lenvatinib or other VEGFR-directed therapy, including sorafenib. * Known metastasis to central nervous system. * Females who are pregnant or breastfeeding. * If \> 1 + proteinuria on urine dipstick testing will undergo 24-hour urine collection for quantitative assessment of proteinuria. Participants with urine protein ≥1g/24 h will be ineligible. * Gastrointestinal malabsorption or any other condition that in the opinion of the investigator might affect the absorption of study drug. * Active infection requiring treatment. * Significant cardiovascular impairment: history of congestive heart failure greater than New York Heart Association (NYHA) Class II, unstable angina, myocardial infarction, or stroke within 6 months of the first dose of study drug, or cardiac arrhythmia requiring medical treatment. * Prolongation of corrected QT interval (QTc) to \> 480 ms as demonstrated by a repeated ECG or any clinically significant ECG abnormality * History of allergic reactions attributed to compounds of similar chemical or biologic composition to lenvatinib. * Any medical or other condition that in the opinion of the investigators would preclude participant's participation in a clinical study.