Inclusion Criteria:

          -  Ability to understand and the willingness to sign a written informed consent document

          -  Histologically (or cytologically) confirmed diagnosis of metastatic, radioiodine (RAI)
             refractory, BRAFV600E/M mutant differentiated thyroid cancer (DTC)

               -  Note: RAI refractoriness is defined as:

                    -  The absence of uptake of RAI on either a low-dose diagnostic whole body
                       scan, or a post-treatment RAI scan in measurable lesions

                    -  Radiographic progression of disease within 12 months of the last course of
                       RAI treatment, or

                    -  Having a cumulative lifetime administered dose of > 600 mCi of RAI

          -  Measurable disease meeting the following criteria and confirmed by radiography review:

               -  At least 1 lesion of >= 1.0 cm in the longest diameter for a non-lymph node or >
                  1.5 cm in the short-axis diameter for a lymph node metastasis that is serially
                  measurable according to Response Evaluation Criteria in Solid Tumors (RECIST) 1.1
                  using computed tomography/magnetic resonance imaging (CT/MRI). If there is only 1
                  target lesion and it is a non-lymph node, it should have a longest diameter of >=
                  1.5 cm

               -  Lesions that have had external beam radiotherapy or locoregional therapies such
                  as radiofrequency (RF) ablation must show evidence of progressive disease based
                  on RECIST 1.1 to be deemed a target lesion.

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 1

          -  Absolute neutrophil count (ANC) >= 1.5 x 10^9/L

          -  Hemoglobin (Hgb) >= 9 g/dL

          -  Platelets (PLT) >= 75 x 10^9/L

          -  Aspartate aminotransferase (AST) and/or alanine aminotransferase (ALT) =< 2.5 x upper
             limit of normal (ULN); participants with liver metastases =< 5 x ULN

          -  Total bilirubin =< 1.5 x ULN

               -  Note: Individuals who have a total bilirubin level > 1.5 x ULN will be allowed if
                  their indirect bilirubin level is =< 1.5 x ULN (i.e., participants with suspected
                  or known diagnosis of Gilbert?s syndrome)

          -  Serum creatinine =< 1.5 x ULN, or calculated creatinine clearance (determined as per
             Cockcroft-Gault) >= 40 mL/min at screening

          -  Left ventricular ejection fraction (LVEF) >= 50% as determined by a multigated
             acquisition (MUGA) scan or echocardiogram

          -  Triplicate average baseline corrected QT (QTc) interval =< 480 ms

          -  Participants of childbearing potential must have a negative serum or urine pregnancy
             test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin
             [HCG]) within 72 hours prior to the start of assigned study intervention

          -  Participants of child-bearing potential agree to use highly effective methods of
             contraception starting with the first dose of assigned study intervention through 6
             months after the last dose of study therapy

          -  Participants of childbearing potential are those who are not proven postmenopausal.
             Postmenopausal is defined as any of the following:

               -  Amenorrheic for 1 year or more following cessation of exogenous hormonal
                  treatments

               -  Luteinizing hormone (LH) and follicle stimulating hormone (FSH) levels in the
                  post-menopausal range for women under 50

               -  Radiation-induced oophorectomy with last menses > 1 year ago

               -  Chemotherapy-induced menopause with > 1 year interval since last menses

               -  Surgical sterilization (bilateral oophorectomy or hysterectomy)

          -  Male participants must agree to use an adequate method of contraception starting with
             the first dose of study therapy through 6 months after the last dose of study therapy

        Exclusion Criteria:

          -  Currently participating and receiving study therapy or has participated in a study of
             an investigational agent and received study therapy or used an investigational
             agent/device within 4 weeks of first dose of study intervention

               -  Note: Individuals in the follow-up phase of a prior investigational study may
                  participate as long as it has been 4 weeks since last dose of the previous
                  investigational agent of device

          -  Participants with active, known, or suspected autoimmune disease. Participants with
             type I diabetes mellitus, residual hypothyroidism due to autoimmune thyroiditis only
             requiring hormone replacement, skin disorders (such as vitiligo, psoriasis, or
             alopecia) not requiring systemic treatment are permitted to enroll

          -  Prior treatment with selective/potent BRAF/MEK inhibitors including vemurafenib,
             dabrafenib, encorafenib, selumetinib, trametinib, cobimetinib, binimetinib.

               -  Note: Prior therapy with oral multikinase inhibitors (e.g., lenvatinib,
                  sorafenib, cabozantinib, pazopanib, sunitinib, etc.) remain eligible for study
                  participation.

          -  Prior treatment with anti-PD-1, anti-PD-L1, anti-PD-L2, anti-CD137, or anti-CTLA-4
             antibody (including ipilimumab or any other antibody or drug specifically targeting T
             cell co-stimulation or immune checkpoint pathways)

          -  Participants with a condition requiring systemic treatment with either corticosteroids
             (>= 10 mg daily prednisone or equivalent) or other immunosuppressive medications
             within 14 days of study drug administration. Inhaled or topical steroids are permitted
             in the absence of active autoimmune disease

          -  History of allergy or hypersensitivity to any monoclonal antibody

          -  History or current evidence of retinal vein occlusion (RVO) or current risk factors to
             RVO (e.g. uncontrolled glaucoma or ocular hypertension, history of hyperviscosity or
             hypercoagulability syndromes); history of retinal degenerative disease

          -  Previous or concurrent malignancy within 3 years of study entry, with the following
             exceptions:

               -  Adequately treated basal or squamous cell skin cancer, superficial bladder
                  cancer, prostate intraepithelial neoplasm, carcinoma in-situ of the cervix, or
                  other noninvasive or indolent malignancy; OR

               -  Other solid tumors treated curatively in which the expected rate of recurrence
                  within 5 years is < 5%

          -  Impaired cardiovascular function or clinically significant cardiovascular diseases,
             including any of the following:

               -  History of acute coronary syndromes (including myocardial infarction, unstable
                  angina, coronary artery bypass grafting, coronary angioplasty, or stenting) < 6
                  months prior to screening

               -  Symptomatic congestive heart failure (i.e. grade 2 or higher), history or current
                  evidence of clinically significant cardiac arrhythmia and/or conduction
                  abnormality < 6 months prior to screening. Exceptions include
                  asymptomatic/well-controlled atrial fibrillation/flutter or paroxysmal
                  supraventricular tachycardia

               -  Uncontrolled hypertension defined as persistent elevation of systolic blood
                  pressure >= 150 mmHg or diastolic blood pressure >= 100 mmHg, despite medical
                  therapy

          -  History of thromboembolic or cerebrovascular events =< 12 weeks prior to the first
             dose of study treatment. Examples include transient ischemic attacks, cerebrovascular
             accidents, hemodynamically significant (i.e., massive or sub-massive) deep vein
             thrombosis or pulmonary emboli

               -  Note: Individuals with either deep vein thrombosis or pulmonary emboli that does
                  not result in hemodynamic instability are allowed to enroll as long as they are
                  on a stable dose of anticoagulants for at least 4 weeks

               -  Note: Individuals with thromboembolic events related to indwelling catheters or
                  other procedures may be enrolled

          -  Known positive serology for HIV (human immunodeficiency virus), active hepatitis B,
             and/or active hepatitis C infection

          -  Known history of acute or chronic pancreatitis

          -  Impaired gastrointestinal function or disease that may significantly alter the
             absorption of encorafenib or binimetinib (e.g., uncontrolled vomiting or malabsorption
             syndrome)

          -  Concurrent neuromuscular disorder that is associated with the potential of elevated
             creatine kinase (CK) (e.g., inflammatory myopathies, muscular dystrophy, amyotrophic
             lateral sclerosis, spinal muscular atrophy)

          -  Current use of a prohibited medication (including herbal medications, supplements, or
             foods), as described, or use of a prohibited medication =< 1 week prior to the start
             of study treatment

          -  Any other condition that would, in the investigator?s judgment, contraindicate an
             individual?s participation in the clinical study due to safety concerns or compliance
             with clinical study procedures, e.g., infection/inflammation, intestinal obstruction,
             unable to swallow medication, social/ psychological issues, etc

          -  Participants who have undergone major surgery (e.g., intracranial, intrathoracic, or
             intra-abdominal surgery) =< 3 weeks prior to starting study drug or who have not
             recovered from side effects of such procedure

          -  Participants that are pregnant or nursing (lactating)

          -  Prisoners or individuals who are involuntarily incarcerated

          -  Medical, psychiatric, cognitive or other conditions that may compromise the
             participant?s ability to understand the patient information, give informed consent,
             comply with the study protocol or complete the study