Inclusion Criteria:

          -  • Patients must have one of the following histologies with molecularly-confirmed
             diagnosis that is recurrent or progressive. Patients enrolled will be stratified as
             follows:

               -  Phase I:

                    -  Stratum 1 LGG or HGG with BRAF V600E/D/K mutation

                    -  Stratum 2 LGG with BRAF duplication or fusion with any partner or LGG with
                       neurofibromatosis type 1

               -  Phase II:

                    -  Stratum 3 LGG with BRAF V600E/D/K mutation

                    -  Stratum 4 HGG with BRAF V600E/D/K mutation

                    -  Stratum 5 LGG with BRAF duplication or fusion with any partner

                    -  Stratum 6 LGG with neurofibromatosis type 1

               -  BRAF alterations will be locally determined using molecular methods in a Clinical
                  Laboratory Improvement Act (CLIA)-certified laboratory. Immunohistochemistry for
                  BRAF V600E alone is not adequate and must be confirmed molecularly

                    -  Phase II patients must have bi-dimensionally measurable disease defined as
                       at least one lesion that can be accurately measured in at least two planes.
                       A target lesion should be chosen

                    -  Patients are required to have weight >= 9 kg to enroll on any stratum in the
                       Phase I or Phase II

               -  Phase I only

                    -  Patients enrolled on the 8 mg/kg/day (dose level 1) must have a weight < 90
                       kg

                    -  Patients enrolled on the 15 mg/kg/day (dose level 2) must have a weight < 80
                       kg

                    -  Patients enrolled on the 20 mg/kg/day (dose level 3) must have a weight < 68
                       kg

                         -  Patients must have received prior therapy other than surgery and must
                            have fully recovered from the acute treatment related toxicities
                            (defined as =< grade 1) of all prior chemotherapy, immunotherapy,
                            radiotherapy or any other treatment modality prior to entering this
                            study

                         -  Only applicable to LGG patients on Phase I and all patients on Phase II

               -  Patients must have received prior RAF and/or MEK inhibitor therapy and meet one
                  of the following criteria:

                    -  Did not experience an objective response (defined as < PR) OR

                    -  Achieved an objective response (CR or PR) but progressed while on active
                       therapy

               -  HGG patients on Phase I: may be enrolled regardless of prior MEK /RAF treatment

                  • Imaging must be available for central review to confirm eligibility for LGG
                  patients on the Phase I study and all patients on the Phase II study

               -  Patients with HGG on the phase I study do not require central imaging review for
                  eligibility

               -  Patients with LGG on the Phase I study will not require real-time central imaging
                  review, but imaging must be available for retrospective review in case the
                  subject was enrolled at the RP2D and may be counted as part of the phase II study

                    -  Patients must have received their last dose of known myelosuppressive
                       anticancer therapy at least 21 days prior to enrollment or at least 42 days
                       if nitrosourea

                    -  Patient must have recovered from any acute toxicity potentially related to
                       the agent and received their last dose of the investigational or biologic
                       agent >= 7 days prior to study enrollment. For biologic agents or monoclonal
                       antibodies with a prolonged half-life, at least three half-lives must have
                       elapsed prior to enrollment

                    -  Patients must have had their last fraction of:

               -  Craniospinal irradiation, whole brain radiation, total body irradiation or
                  radiation to >= 50% of pelvis or spine >= 6 weeks (42 days) prior to enrollment

                  ** Focal irradiation >= 14 days prior to enrollment

                    -  Patients with neurological deficits should have deficits that are stable for
                       a minimum of 7 days prior to enrollment.

                    -  Patients with seizure disorders may be enrolled if seizures are controlled.
                       Patients may take non-enzyme inducing anti-epileptic medications

                    -  Patients who are receiving dexamethasone must be on a stable or decreasing
                       dose for at least 1 week prior to enrollment

                    -  Karnofsky performance scale (KPS for > 16 years of age) or Lansky
                       performance score (LPS for =< 16 years of age) assessed within 7 days of
                       enrollment must be >= 50

               -  Patients who are unable to walk because of neurologic deficits, but who are up in
                  a wheelchair, will be considered ambulatory for assessing the performance score

                    -  Absolute neutrophil count >= 1.0 x 10^9 cells/ L

                    -  Platelets >= 100 x 10^9 cells/ L (unsupported, defined as no platelet
                       transfusion within 7 days)

                    -  Hemoglobin >= 8 g/dl (may receive transfusions)

                    -  Total bilirubin =< 1.5 times institutional upper limit of normal (ULN)

                    -  Alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase
                       [SGPT]) < 3 x institutional upper limit of normal (ULN)

                    -  Albumin >= 3 g/dl

                    -  Serum creatinine based on age/gender. Patients that do not meet these
                       criteria but have a 24-hour creatinine clearance or glomerular filtration
                       rate (GFR) (radioisotope or iothalamate) >= 70 mL/min/1.73 m^2 are eligible

               -  Age: 1 to < 2 years; maximum serum creatinine (mg/dL): 0.6 (male); 0.6 (female)

               -  Age: 2 to < 6 years; maximum serum creatinine (mg/dL): 0.8 (male); 0.8 (female)

               -  Age: 6 to < 10 years; maximum serum creatinine (mg/dL): 1 (male); 1 (female)

               -  Age: 10 to < 13 years; maximum serum creatinine (mg/dL): 1.2 (male); 1.2 (female)

               -  Age: 13 to < 16 years; maximum serum creatinine (mg/dL): 1.5 (male); 1.4 (female)

               -  Age: >= 16 years; maximum serum creatinine (mg/dL): 1.7 (male); 1.4 (female)

                    -  Left ventricular ejection fraction greater than the institutional lower
                       limit of normal by echo (while not receiving medications for cardiac
                       function)

                    -  Corrected QT (QTc) =< 480 msec

                    -  Female patients of childbearing potential must have a negative serum or
                       urine pregnancy test within 72 hours prior to receiving the first dose of
                       study medication. If the urine test is positive or cannot be confirmed as
                       negative, a serum pregnancy test will be required

                    -  Females of child-bearing potential must use a highly effective method of
                       contraception during dosing of study treatment and for 16 weeks after
                       stopping study medication.

                    -  Sexually active males must use a condom during intercourse while on study
                       and for 16 weeks after stopping study treatment and agree not to father a
                       child during this period

                    -  The patient or parent/guardian is able to understand the consent and is
                       willing to sign a written informed consent document according to
                       institutional guidelines

        Exclusion Criteria:

          -  • Breast-feeding women are excluded from this study due to risks of fetal and
             teratogenic adverse events as seen in animal/human studies

               -  Patients with any clinically significant unrelated systemic illness (serious
                  infections or significant cardiac, pulmonary, hepatic or other organ
                  dysfunction), that in the opinion of the investigator would compromise the
                  patient's ability to tolerate protocol therapy, put them at additional risk for
                  toxicity or would interfere with the study procedures or results:

                    -  Patients with a prior or concurrent malignancy whose natural history or
                       treatment does not have the potential to interfere with the safety or
                       efficacy assessment of the investigational regimen are eligible for this
                       trial. Patients with NF1 and history of plexiform neurofibroma will be
                       permitted to enroll

                    -  Patients with a previously documented retinal vein occlusion or severe
                       retinopathy

                    -  Presence of active gastrointestinal (GI) disease or other condition (e.g.,
                       small bowel or large bowel resection) that will interfere significantly with
                       the absorption of drugs

               -  Patients who are unable to discontinue prohibited medications or herbal
                  preparations within 7 days of enrollment and 14 days of starting study therapy

               -  Patients who are receiving any other anti-cancer or investigational drug therapy
                  are ineligible

               -  Patients with a history of a known hypersensitivity to dabrafenib, trametinib,
                  HCQ, or any of their excipients or compounds of similar chemical or biologic
                  composition

               -  Prisoners will be excluded from this study.

               -  Patients who in the opinion of the investigator are unwilling or unable to return
                  for required follow-up visits or obtain follow-up studies required to assess
                  toxicity to therapy or to adhere to drug administration plan, other study
                  procedures, and study restrictions