MUC1-Activated T Cells for the Treatment of Relapsed and Resistant Ovarian Cancer

Inclusion Criteria: * PRE-REGISTRATION: Age ≥ 18 years * PRE-REGISTRATION: Diagnosis or history of epithelial ovarian, fallopian tube, carcinosarcoma, or primary peritoneal cancer * PRE-REGISTRATION: Patients must have measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) criteria on study entry, which must include at least 1 lesion that has a single diameter of ≥ 1 cm measured by CT or MRI or the CT portion of the PET/CT * Skin lesions can be used if the area is ≥ 1cm in at least one diameter and measured with a ruler * PRE-REGISTRATION: Relapsed or refractory ovarian cancer previously treated with or intolerant to at least one prior line of therapy with platinum chemotherapy and be relapsed or have tumor evaluable for response if in first line setting resistant or ineligible to platinum. Patients with BRCA1/2 mutations must have received prior treatment with a poly (ADP-ribose) polymerase (PARP) inhibitor to be eligible. Platinum-resistance is defined as any of the following occurring \< 183 days after the last dose of platinum-based chemotherapy: * Development of measurable disease (per RECIST 1.1) * Progression of radiographic disease (per RECIST 1.1) * Increase in CA-125 level to ≥ 2 x upper limit of normal (ULN) (if within normal limits \[WNL\] at the completion of platinum-based chemotherapy) * Increase in CA-125 level to ≥ 2 x nadir (if nadir \> ULN) * If CA-125 is used to determine the date of progression then it must be confirmed by a second CA-125 value ≥ 7 days after the first level and concurrent with imaging changes. The date of the first qualifying CA-125 is used to compute the platinum-free interval * PRE-REGISTRATION: Provide written informed consent * PRE-REGISTRATION: Willingness to provide mandatory blood specimens and biopsy tissue for correlative research * REGISTRATION: Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * REGISTRATION: Histologically confirmed surgical diagnosis of epithelial ovarian, fallopian tube, carcinosarcoma, or primary peritoneal cancer with measurable disease. NOTE: Histologic confirmation of the primary tumor is required. Eligible histologies include serous, endometrioid, clear cell, mucinous, transitional cell, undifferentiated, or mixed carcinoma * REGISTRATION: MUC1 expression in ovarian cancer tumor cells verified by immunohistochemistry (IHC) in a Clinical Laboratory Improvement Act (CLIA) laboratory. Heterogeneous tumor expression of MUC1 is acceptable. MUC1 expression by staining score greater than 0 is deemed positive for this study * REGISTRATION: Expected survival unless investigational therapy is effective is greater than 6 months but less than 24 months * REGISTRATION: Willingness and ability to provide written informed consent * REGISTRATION: Willing to return to Mayo Clinic in Arizona (MCA) for follow-up during the active monitoring phase of the study * REGISTRATION: Willing to undergo leukapheresis for blood component collection * REGISTRATION: Absolute neutrophil count (ANC) ≥ 1500/mm\^3 (performed ≤ 14 days prior to registration) * REGISTRATION: Lymphocyte count ≥ 1500/mm\^3 (performed ≤ 14 days prior to registration) * REGISTRATION: Hemoglobin ≥ 8.0 g/dL (performed ≤ 14 days prior to registration) * REGISTRATION: Platelet count ≥ 30,000/mm\^3 (performed ≤ 14 days prior to registration) * REGISTRATION: Total bilirubin ≤ 2.0 mg/dL unless patient has documented Gilbert's syndrome (subjects with Gilbert's syndrome may be included if their total bilirubin is ≤ 3.0 x ULN and direct bilirubin ≤ 1.5 x ULN) (performed ≤ 14 days prior to registration) * Alanine aminotransferase (ALT) and aspartate amino transferase (AST) ≤ 3 x ULN (≤ 5 x ULN for patients with liver involvement of their cancer) (performed ≤ 14 days prior to registration) * REGISTRATION: Prothrombin time (PT), international normalized ratio (INR) and activated partial thromboplastin time (aPTT) ≤ 1.5 x ULN OR if patient is receiving anticoagulation therapy and INR or aPTT is within target range of therapy (for patients receiving anticoagulation, there should be no prior history of bleeding and no recent deep vein thrombosis \[DVT\]/pulmonary embolism \[PE\] ≤ 6 months prior to registration) (performed ≤ 14 days prior to registration) * REGISTRATION: Calculated creatinine clearance ≥ 30 ml/min using the Cockcroft-Gault formula (performed ≤ 14 days prior to registration) * REGISTRATION: Baseline oxygen saturation ≥ 90% on room air * REGISTRATION: Negative urine or serological pregnancy test ≤ 7 days prior to registration Exclusion Criteria: * Clinically unresolved central nervous system (CNS) metastases. NOTE: Patients with a prior history of brain metastases are allowed if focally treated, radiographically stable for \> 30 days, and not requiring steroid therapy for \> 14 days * Prior treatment targeting MUC1 * Subjects with known plasma cell leukemia (PCL) * Any of the following are excluded because this study involves an agent (CTX) that has known genotoxic, mutagenic and/or teratogenic effects: * Pregnant persons * Nursing persons * Persons of childbearing potential who are unwilling to employ adequate birth control measures * History of myocardial infarction ≤ 6 months prior to registration, and/or congestive heart failure requiring ongoing treatment such as medications and/or an implanted defibrillator to control life-threatening arrhythmias * Failure to recover to grade 1 or baseline from acute, reversible effects of prior therapy regardless of interval since last treatment. EXCEPTION: Grade 2 peripheral (sensory) neuropathy that has been stable for at least 3 months since completion of prior treatment. * Uncontrolled concurrent illness including, but not limited to: * Inability to clear an ongoing or active infection * Symptomatic congestive heart failure * Unstable angina pectoris * Uncontrolled psychiatric problems * Inability to have a caregiver for active oversight during treat