Study of Targeted Therapy vs. Chemotherapy in Patients With Thyroid Cancer

Inclusion Criteria: * Patient must be ≥ 18 years of age * Patient must have an Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * Patient must have differentiated thyroid cancer (DTC) with BRAF V600E mutation as determined by local testing, including the following subtypes (Note: results of a previous biopsy will be accepted): * Papillary thyroid carcinoma including histological variants of papillary thyroid carcinoma (PTC) such as follicular variant, tall cell, columnar cell, cribriform-morular, solid, oxyphil, Warthin-like, trabecular, tumor with nodular fasciitis-like stroma, Hürthle cell variant of papillary carcinoma, poorly differentiated. * Follicular thyroid carcinoma including histological variants of follicular thyroid carcinoma (FTC) such as Hürthle cell, clear cell, insular, and poorly differentiated * Patient must have been previously treated with or deemed ineligible for treatment with Iodine-131 for DTC, and must be receiving thyroxine suppression therapy * Patient must have had prior treatment with at least one of the following vascular endothelial growth factor receptors (VEGFR)-targeting tyrosine kinase inhibitor (TKI) agents for DTC: lenvatinib or sorafenib. * NOTE: Up to two prior VEGFR-targeting TKI agents are allowed including, but not limited to lenvatinib and sorafenib * Patient must have measurable disease according to Response Evaluation Criteria in Solid Tumors (RECIST) 1·1 on chest CT (computed tomography)/abdominal/pelvis CT/MRI (magnetic resonance imaging) performed within 4 weeks prior to randomization * Patient must have radiographic progression by RECIST 1.1 over any time interval on or after most recent prior systemic treatment * Patient must not have any of the following cardiovascular and thromboembolic disorders or medical conditions: * Congestive heart failure class 3 or 4 as defined by the New York Heart Association, unstable angina pectoris, or serious cardiac arrhythmias. * Uncontrolled hypertension defined as sustained blood pressure \> 150 mm Hg systolic or \> 100 mm Hg diastolic despite optimal antihypertensive treatment. * Stroke, myocardial infarction, or thromboembolic event (e.g., deep venous thrombosis, pulmonary embolism) within 6 months prior to randomization. Patients with more recent diagnosis of deep venous thrombosis are allowed if stable and treated with therapeutic anticoagulation for at least 6 weeks prior to randomization * Patient must not have any clinically significant hematemesis or haemoptysis of \> 0·5 teaspoon (\> 2·5 mL) of red blood or history of other significant bleeding within 3 months prior to randomization * Patient must not have any cavitating pulmonary lesion(s) or lesions invading major pulmonary blood vessels * Patient must not be on any concomitant anticoagulation with oral anticoagulants or platelet inhibitors, except for the following allowed agents: * Low-dose aspirin for cardioprotection. * Therapeutic anticoagulation with any agent in patients (1) without known brain metastases, (2) on a stable dose for at least 6 weeks prior to randomization, and (3) with no clinically significant hemorrhagic complications from the anticoagulation regimen or the tumor * Patient must not have any gastrointestinal (GI) disorders associated with a high risk of perforation or fistula formation: * Tumors invading the GI tract, active peptic ulcer disease, inflammatory bowel disease, ulcerative colitis, diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis, acute pancreatitis, or acute obstruction of the pancreatic or biliary duct, or gastric outlet obstruction * Abdominal fistula, GI perforation, bowel obstruction, or intra-abdominal abscess within 6 months prior to randomization * Patient must have completed any prior local therapy (e.g., surgery, radiation, ablation) at least 4 weeks prior to randomization, with complete wound healing and resolution of clinically relevant complications from prior local therapy * Patient must not have had major surgery (e.g., GI surgery, removal or biopsy of brain metastasis) within 8 weeks prior to randomization. Complete wound healing from major surgery must have occurred 4 weeks prior to randomization and from minor surgery (e.g., simple excision, tooth extraction) at least 10 days prior to randomization * Patient must not have pathologically documented anaplastic histology or transformation on any recurrence specimen collected after initial thyroidectomy/neck dissection * Patient must not have had prior treatment with cabozantinib or any prior BRAF targeted therapy for thyroid cancer * NOTE: Prior treatment with checkpoint inhibitor (e.g., pembrolizumab) for thyroid cancer is allowed, but must be noted on the eCRF (electronic case report form) * Patient must not be pregnant or breast-feeding due to the potential harm to an unborn fetus and possible risk for adverse events in nursing infants with the treatment regimens being used. All patients of childbearing potential must have a blood test or urine study within 14 days prior to randomization to rule out pregnancy. A patient of childbearing potential is defined as someone, regardless of whether they have undergone tubal ligation, who meets the following criteria: 1) has achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy; or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * Patients must not expect to conceive or father children by using accepted and effective method(s) of contraception or by abstaining from sexual intercourse for the duration of their participation in the study and for 2 weeks after the last dose of dabrafenib and 4 months after the last dose of trametinib or cabozantinib. Patients must also not breastfeed while on study treatment and fo