Enhancing Radioiodine Incorporation Into BRAF Mutant Thyroid Cancers With the Combination of Vemurafenib and KTN3379

Inclusion Criteria: * Patients must have histologically or cytologically confirmed thyroid carcinoma of follicular origin (including papillary, follicular, or poorly differentiated subtypes and their respective variants). * Confirmation in a CLIA certified laboratory or in an FDA-approved assay that one of the patient's thyroid tumors (primary tumor, recurrent tumor, or metastasis) possesses a BRAF mutation at V600. * Patients must have measurable disease defined by RECIST criteria 1.1. * 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 * Age ≥ 18 years. * ECOG performance status ≤ 2 * Patients must have normal organ and marrow function as defined below: * Absolute neutrophil count (ANC) \> 1500/mcl * Hemoglobin ≥ 9 g/dL * Platelets ≥ 100,000/mcl * 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 Exclusion Criteria: * Concomitant malignancies or previous malignancies treated within the past 3 years. Exception: Patients who have been disease-free for 3 years, patients with a history of completely resected non-melanoma skin cancer, and/or patients with indolent secondary malignancies, are eligible. * Use of other investigational drugs within 28 days preceding the first dose of vemurafenib on this study. * Symptomatic or untreated leptomeningeal or brain metastases or spinal cord compression. * History or evidence of cardiovascular risk including any of the following: * Corrected QT (QTc) interval ≥ 450 msec at baseline or history of congenital long QT syndrome or uncorrectable electrolyte abnormalities. (Patients with well controlled atrial fibrillation are exempt from this criteria.) * 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.