Inclusion Criteria:

          -  Patients can be previously or simultaneously enrolled in the natural history ECD
             protocol #11-HG-0207, "Clinical and Basic Investigations into Erdheim Chester
             disease". Eligible patients must have been diagnosed with Erdheim Chester disease,
             confirmed by pathological evaluation of the affected tissue with adequate staining.
             Affected tissue must harbor the BRAF V600 mutation

          -  Patients must have measurable or evaluable disease according to Response Evaluation
             Criteria in Solid Tumors 1.1

          -  Prior treatment, involving interferon, anakinra, imatinib, steroids, chemotherapy
             with, but not limited to cladribine, vinblastine, 6-mercaptopurine and etoposide, or
             other medications used empirically for the treatment of ECD, will be acceptable. These
             therapies should have been completed and discontinued 4 weeks or for biologic agents 4
             weeks or 5 half-lives (whichever comes shorter) prior to enrollment in this study

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 60%)

               -  Exception will be made for patients with ECOG performance status =< 3 and
                  Karnofsky performance scale >= 50%, who require the use of wheelchairs, walkers
                  or canes as well as assistance with daily routines secondary to disabilities
                  caused by ECD cerebellar or brain disease that has been stable for >= 3 months

          -  Life expectancy of greater than 3 months

          -  Able to swallow and retain oral medication and must not have any clinically
             significant gastrointestinal abnormalities that may alter absorption such as
             malabsorption syndrome or major resection of the stomach or bowels

          -  Patients must have BRAF V600 mutation in the tumor tissue and/or cell-free
             deoxyribonucleic acid (DNA), identified by a Clinical Laboratory Improvement Act
             (CLIA)-certified lab

          -  Absolute neutrophil count (ANC) >= 1.0 x 10^9/L

          -  Hemoglobin >= 8 g/dL

          -  Platelets >= 75 x 10^9/L

          -  Serum bilirubin =< 1.5 x institutional upper limit of normal (ULN) except subjects
             with known Gilbert's syndrome

          -  Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =< 3 x
             institutional ULN

          -  Serum creatinine =< 1.5 x institutional ULN

          -  Left ventricular ejection fraction >= institutional lower limit of normal (LLN) by
             echocardiogram (ECHO)

          -  Women of childbearing potential must have a negative serum pregnancy test within 14
             days prior to registration or randomization

          -  Pregnancy and breast feeding. The effects of dabrafenib and trametinib on the
             developing human fetus are unknown. For this reason women of child-bearing potential
             must agree to use adequate contraception (barrier method of birth control, or
             abstinence; hormonal contraception is not allowed due to drug-drug interactions which
             can render hormonal contraceptives ineffective) for the duration of study
             participation, and for at least 2 weeks after treatment with dabrafenib or for 4
             months after dabrafenib in combination with trametinib. Should a woman become pregnant
             or suspect she is pregnant while she is participating in this study, she should inform
             her treating physician immediately. Based on studies in animals, it is also known that
             dabrafenib may cause damage to the tissue that makes sperm. This may cause sperm to be
             abnormal in shape and size and could lead to infertility, which may be irreversible.
             Safety and efficacy of the combination of dabrafenib and trametinib in pediatric
             populations have not been investigated. Dabrafenib or trametinib-dabrafenib
             combination should not be administered to pediatric populations outside clinical
             trials

          -  Therapeutic level dosing of warfarin can be used with close monitoring of prothrombin
             time (PT)/international normalized ratio (INR) by the site. Exposure may be decreased
             due to enzyme induction when on treatment, thus warfarin dosing may need to be
             adjusted based upon PT/INR. Consequently, when discontinuing dabrafenib, warfarin
             exposure may be increased and thus close monitoring via PT/INR and warfarin dose
             adjustments must be made as clinically appropriate. Prophylactic low dose warfarin may
             be given to maintain central catheter patency

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

        Exclusion Criteria:

          -  Inability to provide informed consent

          -  Patients treated with prior BRAF and/or MEK inhibitors

          -  Current use of a prohibited medication. Patients receiving any medications or
             substances that are strong inhibitors or inducers of CYP3A or CYP2C8 are ineligible.
             Current use of, or intended ongoing treatment with: herbal remedies (e.g., St. John's
             wort), or strong inhibitors or inducers of P-glycoprotein (Pgp) or breast cancer
             resistance protein 1 (Bcrp1) should also be excluded

          -  Unresolved toxicity of National Cancer Institute Common Terminology Criteria for
             Adverse Events, version 5.0 (NCI CTCAE v5.0) grade 2 or higher from previous
             anti-cancer therapy, except alopecia, at the time of randomization

          -  Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral
             therapy are ineligible because of the potential for pharmacokinetic interactions with
             dabrafenib

          -  A history of hepatitis B virus (HBV) or hepatitis C virus (HCV) infection (with the
             exception of cleared HBV and HCV infection, which will be allowed)

          -  Presence of active malignancy other than the study indication

          -  Patients with known RAS in ECD, or history of other malignancies with RAS mutations

          -  Leptomeningeal or brain metastases are allowed. Subjects on a stable dose of
             corticosteroids can be enrolled with approval of the principal investigator (PI) and
             Cancer Therapy Evaluation Program (CTEP) medical monitor. Subjects must not receive
             enzyme-inducing anticonvulsants

          -  History or evidence of cardiovascular risks, except stable ECD cardiac lesion,
             including any of the following:

               -  QT interval corrected for heart rate using the Bazett's formula (QTcB) >= 480
                  msec

               -  History of acute coronary syndromes (including myocardial infarction or unstable
                  angina), coronary angioplasty, or stenting within the past 24 weeks prior to
                  randomization

               -  History or evidence of current class II, III, or IV heart failure as defined by
                  the New York Heart Association (NYHA) functional classification system

               -  Intra-cardiac defibrillators

               -  Abnormal cardiac valve morphology (>= grade 2) documented by ECHO; (subjects with
                  grade 1 abnormalities [i.e., mild regurgitation/stenosis] can be entered on
                  study). Subjects with moderate valvular thickening should not be entered on study

               -  History or evidence of current clinically significant uncontrolled cardiac
                  arrhythmias; clarification: subjects with atrial fibrillation controlled for > 30
                  days prior to dosing are eligible

               -  Treatment refractory hypertension defined as a blood pressure of systolic > 140
                  mmHg and/or diastolic > 90 mm Hg which cannot be controlled by anti-hypertensive
                  therapy

          -  Known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs
             chemically related to the study treatments, their excipients, and/or dimethyl
             sulfoxide (DMSO)

          -  Any serious or unstable pre-existing medical conditions (aside from malignancy
             exceptions specified above), psychiatric disorders, or other conditions that could
             interfere with the subject's safety, obtaining informed consent, or compliance with
             study procedures

          -  Pregnant women are excluded from this study because of the potential for teratogenic
             or abortifacient effects. Because there is an unknown but potential risk for adverse
             events in nursing infants secondary to treatment of the mother with
             dabrafenib/trametinib, breastfeeding should be discontinued prior to treatment with
             dabrafenib/trametinib. These potential risks may also apply to other agents used in
             this study

          -  History of retinal vein occlusion (RVO)

          -  Interstitial lung disease or pneumonitis not secondary to ECD

          -  Central serous retinopathy (CSR) including presence of predisposing factors to RVO or
             CSR (e.g., uncontrolled glaucoma or ocular hypertension, uncontrolled diabetes
             mellitus, or a history of hyperviscosity or hypercoagulability syndromes); or visible
             pathology (e.g., evidence of optic disc cupping, evidence of new visual field defects
             on automated perimetry, or intraocular pressure > 21 mmHg as measured by tonography)
             as assessed by ophthalmic examination

          -  Inability to travel to the treating center

          -  Patients with confirmed diagnosis of ECD that are asymptomatic and with no visceral
             involvement are not eligible for this trial (patients with no target lesions as per
             RECIST 1.1 criteria)