Encorafenib Plus Binimetinib for People With BRAF V600 Mutated Relapsed/Refractory HCL

* INCLUSION CRITERIA: * Histologically confirmed diagnosis of HCL according to morphological and immunophenotypic criteria of World Health Organization (WHO) classification \[WHO, 2008 revised 2016\] of lymphoid neoplasm. Participants should have at least one of the following indications for therapy: * Absolute neutrophil count (ANC) \<1 x10(3)/mcL * Hemoglobin \<10g/dL * Platelets\<100 x10(3)/mcL * Symptomatic splenomegaly * Enlarging HCL mass \> 2cm in short axis (\>0.5cm in short axis for CNS mass) * Leukemia cell count\>5x10(3)/mcL Participants who have eligible blood counts within 4 weeks prior to initiation of study therapy will not be considered ineligible if subsequent blood counts prior to initiation of study therapy fluctuate and become ineligible up until the time of the initiation of study therapy. * Participants must have BRAF V600 mutation as confirmed from fresh bone marrow aspirate, peripheral blood sample, or lymph node/mass by the Laboratory of Pathology, NCI. This may be done by PCR or sequence-based assays. * Participants who are ineligible for, unable to obtain in a timely manner, cannot access, unwilling to undergo or have failed Moxetumomab Pasudotox trial at NCI * Refractory or relapsed disease- defined as either: * Refractory- no response or disease progression in \<=1 year following first-line treatment with a purine analog, or * Relapsed- having relapsed following treatment with at least 1 prior purine-analog treatment * Age \>=18 years * Eastern Cooperative Oncology Group (ECOG) performance status \<=2 (Karnofsky \>=60%) * Participants must have adequate organ and marrow function as defined below: * Total bilirubin \<= 3x upper limit of normal (ULN), unless consistent with Gilbert s (ratio between total and direct bilirubin \> 5) * AST and ALT \<= 3x ULN * Alkaline phosphatase \< 2.5x ULN * Serum creatinine \<= 1.5 mg/dL or creatinine clearance \>= 60 mL/min/1.73 m\^2 for participants with creatinine levels above institutional normal calculated using estimated glomerular filtration rate (eGFR) * Serum albumin \>= 2 g/dL * Prothrombin time (PT)/International Normalized Ratio (INR) \< 2.5x ULN (If on warfarin, PT/INR \< 3.5x ULN; If on any other anticoagulation, (PT) \< 2.5x ULN * Fibrinogen \>= 0.5x lower limit of normal * Females of childbearing potential (FOCBP) who are sexually active with a non-sterilized male partner must use a highly effective method of contraception and not donate ova prior to study entry and or the duration of study treatment and until 30 days after the last dose of study drug. Periodic abstinence, the rhythm method, and the withdrawal method are not acceptable methods of contraception. There is a potential for encorafenib to induce CYP3A4, which may reduce the effectiveness of hormonal contraception methods. Therefore, the use of at least 1 form of non-hormonal contraception is required for females of childbearing potential during study treatment in this study. Females of childbearing potential are defined as those who are not surgically sterile (i.e., bilateral tubal ligation, bilateral oophorectomy, or complete hysterectomy) or those who are premenarchal or postmenopausal (defined as 12 months with no menses without an alternative medical cause). A highly effective method of contraception is defined as one that results in a low failure rate (i.e., less than 1% per year) when used consistently and correctly. Not all methods of contraception are highly effective. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. --Male participants must use a condom during treatment and through 90 days after the end of systemic exposure to study drug/treatment. If the male participant has a partner that is of child-bearing potential, the partner should also use contraception through 90 days after the end of systemic exposure to study drug/treatment. In addition, male participants must refrain from donating sperm during the study treatment and through 90 days after the end of systemic exposure to study drug/treatment. Males who have had a vasectomy qualify as having met the requirement for a highly effective birth control method. * Nursing participants must be willing to discontinue nursing from study treatment initiation through 14 days after the last dose of the study drugs. * Ability of participant to understand and the willingness to sign a written informed consent document. * Must co-enroll in study 10-C-0066: Collection of Human Samples to Study Hairy Cell and other Leukemias, and to Develop Recombinant Immunotoxins for Cancer Treatment EXCLUSION CRITERIA: * Participants who have had chemotherapy, immunotherapy, investigational agent or radiotherapy within 4 weeks prior to the start of study treatment. * Prior therapy with encorafenib and/or binimetinib * Participants who are receiving any other investigational agents or have received an investigational agent within 14 days prior to the start of study treatment. * Participants who have undergone major surgery \<=6 weeks prior to start of study treatment or who have not recovered from side effects of such procedure * Known hypersensitivity or contraindication to any component of binimetinib or encorafenib or their excipients * Inability to swallow and retain study drugs. * Pregnant women as evaluated by a positive serum or urine beta-human chorionic gonadotropin (beta-hCG). * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, cardiac dysfunction, uncontrolled pulmonary infection, pulmonary edema or psychiatric illness/social situations that would limit compliance with study requirements. * Evidence of active Hepatitis B Virus (HBV) or Hepatitis C Virus (HCV) infection. Note: Participants with laboratory evidence of cleared HBV or HCV infection may be enrolled. If positive for Hepatitis B core antibody or surface antigen the participant must be on Tenofovir or Entecavir and Hepatitis B deoxyribonucleic acid (DNA) viral load must be \<2000 IU/mL * Active second malignancy requiring treatment other than minor resection of indolent cancers like basal cell and squamous skin cancers. * Human immunodeficiency virus (HIV)-positive participants unless taking appropriate anti-HIV medications with a CD4 count of \> 200. Otherwise, there may be an increased risk of infections. * History of an allogeneic bone marrow or stem cell transplant. * Known history of chronic pancreatitis. * Impaired cardiovascular function or clinically significant cardiovascular disease including, but not limited to, any of the following: * History of acute coronary syndromes (including myocardial infarction, unstable angina, coronary artery bypass grafting, coronary angioplasty or stenting) \<3months prior to initiation of study therapy * Congestive heart failure requiring treatment (New York Heart Association Grade greater than or equal to 2); * Left ventricular ejection fraction (LVEF) \< 50% as determined by Multigated Acquisition Scan (MUGA) or Transthoracic echocardiogram (TTE); * Uncontrolled hypertension defined as persistent systolic blood pressure greater than or equal to 160 mmHg or diastolic blood pressure greater than or equal to 100 mmHg despite current therapy; * History or presence of clinically significant cardiac arrhythmias (including resting bradycardia, uncontrolled atrial fibrillation or uncontrolled paroxysmal supraventricular tachycardia); * Triplicate average baseline QTcF interval greater than or equal to 480 ms. * Impairment of gastrointestinal function or disease which may significantly alter the absorption of study drug (e.g., active ulcerative disease, uncontrolled vomiting or diarrhea, malabsorption syndrome, small bowel resection with decreased intestinal absorption), or recent (less than or equal to 3 months) history of a partial or complete bowel obstruction, or other conditions that will interfere significantly with the absorption of oral drugs. * Concurrent neuromuscular disorder that is associated with elevated creatinine kinase CK (e.g., inflammatory myopathies, muscular dystrophy, amyotrophic lateral sclerosis, spinal muscular atrophy). * History or current evidence of retinal vein occlusion (RVO) or current risk factors for RVO (e.g., uncontrolled glaucoma or ocular hypertension, history of hyperviscosity or hypercoagulability syndromes); history of maculopathy or retinopathy for which there is an increased risk of MEK induced exudation (e.g., Central Serous Retinopathy). * History of thromboembolic or cerebrovascular events less than or equal to 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: Participants 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: Participants with thromboembolic events related to indwelling catheters or other procedures may be enrolled * Participants taking strong CYP3A4 inhibitors and strong/moderate CYP3A4 inducers