Lenvatinib 24 mg/Day Versus 10 mg/Day to Treat Symptomatic or Progressive Radioactive Iodine Resistant (RAIR) Differentiated Thyroid Cancer (DTC)

Inclusion Criteria: * Histologically confirmed DTC, defined as papillary, follicular, or Hurthle Cell thyroid cancer. Papillary has several sub-types such as tall-cell and columnar cell, which are all allowed. * Patient must have incurable RAIR DTC, defined as disease not amenable to cure by surgery AND meeting one or more of the following criteria: * one or more sites of disease that do not take up RAI. * disease progression on RAI (given within the last 12 months). * receipt of cumulative dose of RAI of ≥ 600mCi. * patient declines or is ineligible for surgery and/or RAI. * Measurable or evaluable disease per RECIST 1.1. * No more than 1 prior line of VEGF/VEGFR targeted therapy for DTC. Examples of VEGF/VEGFR therapies include sorafenib, pazopanib, vandetinib, axitinib, sunitinib, and cabozantinib, but others exist. * Symptomatic (defined by usual standard of care clinical criteria) or progressive disease on most recent prior treatment (ex: surgery, RAI, or TKI/targeted therapy) by RECIST 1.1 over the last 16 months. * At least 18 years of age. * ECOG performance status ≤ 2. * Screening blood pressure measurement \<140/90. Retesting is allowed. * Adequate bone marrow and organ function as defined below: * Absolute neutrophil count ≥ 1.0 K/cumm * Platelets ≥ 100 K/cumm * Hemoglobin ≥ 9.0 g/dL * Total bilirubin ≤ 1.5 x IULN * AST(SGOT)/ALT(SGPT) ≤ 3.0 x IULN (5.0 x IULN in the presence of hepatic metastases) * Creatinine clearance \> 30 mL/min by Cockcroft-Gault * UPC ≤ 1000 mg/G * QTcF \< 481 msec * The effects of lenvatinib on the developing human fetus are unknown. For this reason, women of childbearing potential and men must agree to use adequate contraception prior to study entry, for the duration of study participation, and for 30 days after the last dose of lenvatinib. Should a woman become pregnant or suspect she is pregnant while participating in this study or should a man suspect he has fathered a child, s/he must inform her treating physician immediately. * Ability to understand and willingness to sign an IRB approved written informed consent document. Legally authorized representatives may sign and give informed consent on behalf of study participants. Exclusion Criteria: * Anaplastic, poorly differentiated/high-grade, and medullary thyroid cancers. * Prior treatment with lenvatinib. * Prior or concurrent malignancy whose natural history has the potential to interfere with the safety or efficacy assessment of the investigational regimen. Patients with prior or concurrent malignancy that does NOT meet that definition are eligible for this trial * Currently receiving any other investigational agents. * Patients with untreated brain metastases. Patients with treated brain metastases are allowed if post-treatment brain-imaging after CNS-directed therapy shows no evidence of progression. * A history of allergic reactions attributed to compounds of similar chemical or biologic composition to lenvatinib or other agents used in the study. * Use of concurrent medications that have a high risk for QTc prolongation. A 7 day washout period of the high-risk medication is required prior to the first dose of Lenvatinib if a patient discontinues the high risk medication for trial enrollment. * Uncontrolled intercurrent illness including, but not limited to: ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, or cardiac arrhythmia. Patients with a 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 Function Classification; to be eligible for this trial, patients should be a class 2B or better. * Pregnant and/or breastfeeding. Women of childbearing potential must have a negative urine pregnancy test within 14 days of C1D1. * HIV-infected if not on effective anti-retroviral therapy with undetectable viral load for 6 months. Patients with HIV who are receiving effective anti-retroviral therapy and have had an undetectable viral load for at least 6 months are eligible. HIV testing not required in the absence of known history of infection. * Evidence of chronic hepatitis B virus (HBV) that is detectable on suppressive therapy. Patients with evidence of chronic HBV infection with undetectable HBV viral load on suppressive therapy are eligible. HBV testing not required in the absence of known history of infection. * History of hepatitis C virus (HCV) infection that has not been cured or that has a detectable viral load. Patients with a history of HCV that has been treated and cured are eligible. Patients with HCV infection who are currently on treatment and have an undetectable HCV viral load are eligible. HCV testing not required in the absence of known history of infection. * Major surgery ≤ 14 days prior to C1D1; any surgical wound must be fully healed prior to C1D1. Inclusion Criteria for Crossover: * Measurable or evaluable disease per RECIST 1.1. * Symptomatic (defined by usual standard of care clinical criteria) or progressive disease by RECIST 1.1 while on lenvatinib. * ECOG performance status ≤ 2. * Re-screening blood pressure measurement \<140/90. * Adequate bone marrow and organ function as defined below: * Absolute neutrophil count ≥ 1.0 K/cumm * Platelets ≥ 100 K/cumm * Hemoglobin ≥ 9.0 g/dL * Total bilirubin ≤ 1.5 x IULN * AST(SGOT)/ALT(SGPT) ≤ 3.0 x IULN (5.0 x IULN in the presence of hepatic metastases) * Creatinine clearance \> 30 mL/min by Cockcroft-Gault * UPC ≤ 1000 mg/G * QTcF \< 481 msec * The most recent dose level of lenvatinib must be 10 mg/day. Exclusion Criteria for Crossover: * Currently receiving any other investigational agents. * Patients with untreated brain metastases. Patients with treated brain metastases (including those who had intracranial progression while on Lenvatinib) are allowed if post-treatment brain-imaging after CNS-directed therapy shows no evidence of progression. * Use of concurrent medications that have a high risk for QTc prolongation. * Uncontrolled intercurrent illness including, but not limited to: ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, or cardiac arrhythmia. Patients with a 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 Function Classification; to be eligible for this trial, patients should be a class 2B or better. * Experienced a grade three or higher treatment related adverse event requiring a dose delay on the 10 mg/day lenvatinib cohort * Pregnant and/or breastfeeding.