Combination of Trametinib (MEK Inhibitor) and Hydroxychloroquine (HCQ) (Autophagy Inhibitor) in Patients With KRAS Mutation Refractory Bile Tract Carcinoma (BTC).

* INCLUSION CRITERIA: * Histopathological confirmation of * biliary tract carcinoma (BTC) OR carcinoma in the setting of clinical and radiological characteristics which, together with the pathology, are highly suggestive of a diagnosis of BTC Note: The term BTC includes intra- or extra- hepatic cholangiocarcinoma (CCA), gallbladder cancer or ampullary cancer. * The tumor must have Kirsten rat sarcoma (KRAS) mutation(s) of clinical significance, confirmed by National Cancer Institute (NCI) Laboratory of Pathology or by Food and Drug Administration (FDA). * Patients must have received or been intolerant of at least one line of chemotherapy. * Patients must have at least 1 measurable lesion by Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1 * Patients must have disease that is not amenable to potentially curative resection, ablation or transplantation. * Age greater than or equal to 18 years. * Performance status Eastern Cooperative Oncology Group (ECOG) 0-2 * If liver cirrhosis is present, patient must have a Child-Pugh score \<7 (Class A) * Patients must have adequate organ and marrow function as defined below: * absolute neutrophil count (ANC) greater than or equal to 1,500/mcL * platelets greater than or equal to 100,000/mcL * hemoglobin greater than or equal to 9 g/dL * total bilirubin if cirrhosis present: Part of Child Pugh requirement. If no cirrhosis: bilirubin should be less than or equal to 1.5 x upper limit of normal (ULN) * alanine aminotransferase (ALT) or aspartate aminotransferase (AST) less than or equal to 5 x ULN. * Creatinine OR Measured or calculated creatinine clearance (CrCl) (eGFR may also be used in place of CrCl)\*\*\* : \< 1.5x institution upper limit of normal OR greater than or equal to 30 mL/min/1.73 m\^2 for participant with creatinine levels greater than or equal to 1.5 X institutional ULN Notes: ALT (SGPT)=alanine aminotransferase (serum glutamic pyruvic transaminase); AST (SGOT)=aspartate aminotransferase (serum glutamic oxaloacetic transaminase); GFR=glomerular filtration rate; ULN=upper limit of normal. \*\*\*Creatinine clearance (CrCl) or estimated glomerular filtration rate (eGFR) should be calculated per institutional standard. * Patients must have at least 1 focus of disease that is amenable to mandatory tumor biopsies and be willing to undergo this. Ideally, the biopsied lesion should not be one of the target measurable lesions, although this can be up to the discretion of the investigators. * The study drugs are harmful for developing human fetus. For this reason, women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) at the study entry, for the duration of study treatment and up to 4 months after the last dose of the study drug(s). 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. * Patients must be able to understand and be willing to sign a written informed consent. EXCLUSION CRITERIA: * Patients who have had standard-of-care anti-cancer therapy within 2 weeks of treatment initiation or therapy with investigational agents (e.g., chemotherapy, immunotherapy, endocrine therapy, targeted therapy, biologic therapy, tumor embolization, monoclonal antibodies or other investigation agents), large field radiotherapy, or major surgery within 4 weeks of treatment initiation. * Any unresolved toxicity NCI Common Terminology Criteria for Adverse Events (CTCAE) v.5 Grade greater than or equal to 2 from previous anticancer therapy with the exception of alopecia, vitiligo, and the laboratory values defined in the inclusion criteria. Patients with Grade greater than or equal to 2 neuropathy will be evaluated on a case-by-case basis. * Has biliary duct obstruction, unless a treatable, clinically relevant obstruction has been relieved by internal endoscopic drainage/stenting, palliative by-pass surgery or percutaneous drainage prior to treatment initiation. * Patients with known brain metastases are excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. * Patients with signs of liver failure, e.g., clinically significant ascites, encephalopathy, or variceal bleeding within six months before treatment initiation. * History or current evidence of retinal vein occlusion (RVO) or current risk factors for RVO (e.g., uncontrolled glaucoma or ocular hypertension, history of hyper viscosity or hypercoagulability syndromes) * Current evidence of uncontrolled, significant intercurrent illness including, but not limited to, the following conditions: * Cardiovascular disorders: Congestive heart failure New York Heart Association class 3 or 4, unstable angina pectoris, serious cardiac arrhythmias, stroke (including transient ischemic attack \[TIA\]), myocardial infarction (MI), or other ischemic event, or thromboembolic event (e.g., deep venous thrombosis, pulmonary embolism) within 3 months before treatment initiation * History of glucose-6-phosphate dehydrogenase (G6PD) deficiency * History of seizures * Patients who are planning on embarking on a new strenuous exercise regimen after first dose of study treatment. Muscular activities, such as strenuous exercise, that can result in significant increases in plasma creatine kinase (CK) levels should be avoided while on study treatment * Patients who have neuromuscular disorders that are associated with elevated CK (e.g., inflammatory myopathies, muscular dystrophy, amyotrophic lateral sclerosis, spinal muscular atrophy) * Impairment of gastrointestinal function or gastrointestinal disease (e.g., ulcerative disease, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome, or small bowel resection that under the judgment of the principal investigator (PI) may impair absorption of study drugs) * Any other condition that would, in the Investigator s judgment, contraindicate the patient 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 (patients may not receive drug through a feeding tube), social/psychological issues, etc. * Screening corrected QT interval by Fridericia's (QTcF) \> 500 msec * Known infection with human immunodeficiency virus (HIV), unless patient is on effective anti-retroviral therapy with undetectable viral load within 6 months of treatment initiation * Known chronic hepatitis B virus, unless hepatitis B virus (HBV) viral load is undetectable. * Known history of hepatitis C virus (HCV) infection, unless completed treatment and cured with undetectable HCV viral load. * Known prior severe hypersensitivity to study drugs or any component in its formulations (CTCAE v5.0 grade \>= 3). * Pregnant women are excluded from this study because study therapy can cause fetal harm. Because there is potential risk for adverse events in nursing infants secondary to treatment of the mother with study therapy, breastfeeding should be discontinued if the mother is treated with study drugs.