Autologous huMNC2-CAR44 or huMNC2-CAR22 T Cells for Breast Cancer Targeting Cleaved Form of MUC1 (MUC1*)

Please note that results of tests and/or procedures conducted as per standard of care purposes may be used for research purposes if conducted within the protocol-defined window prior to screening/leukapheresis and/or T-Cell Therapy. Inclusion Criteria: 1. Confirmation of diagnosis of breast cancer by pathology review of initial or subsequent biopsy or other pathologic material at the City of Hope Pathology department. ER, PR, and HER2 status known and documented per ASCO/CAP guidelines. 1. For dose expansion cohorts, tumors with ER and/or PR ≥1% will be considered hormone receptor positive. Tumors with ER and PR \<1% will be considered hormone receptor negative. HER2 status will be determined by IHC or FISH per ASCO/CAP guidelines. Patients will be allocated to expansion cohorts according to guidelines in table below. 2. Dose expansion cohorts Expansion Cohort Hormone Receptor status HER2 status Luminal ER and/or PR \>/=1% positive Negative by IHC or FISH HER2 positive Any ER or PR status Positive by IHC or FISH Triple Negative ER and PR \<1% Negative by IHC or FISH 2. Patients must have received standard metastatic systemic therapy per NCCN guidelines or institutional practice which are known to confer benefit. No maximum on number of prior systemic treatment regimens. 1. Patients with hormone receptor positive disease must have received at least 3 prior endocrine therapies and at least 2 prior lines of chemotherapy in the metastatic setting. 2. Patients with HER2 positive breast cancer must have received at least 3 prior HER2- directed therapies (trastuzumab, pertuzumab, TDM-1 or others) in the metastatic setting. 3. Patients with triple negative disease must have received at least 2 prior lines of chemotherapy in the metastatic setting. 3. MUC1\* membrane expression ≥30% by immunohistochemistry on a tumor specimen obtained at screening or previous tumor specimen that is less than 6-months old (see Appendix I for examples of MUC1\* expression patterns). 4. Patients must be 18 years of age or older, of any gender, race or ethnicity. 5. Patients must be capable of understanding and providing a written informed consent. 6. Patients must have a Karnofsky performance status of ≥60%. 7. Patients must have measurable disease by at least one of the criteria below: 1. Extra skeletal disease that can be accurately measured by CT or MRI per RECIST 1.1, 2. Skeletal or bone-only metastases measurable by FDG PET imaging. 8. Negative serum pregnancy test within 14 days of planned leukapheresis and within 28 days of lymphodepleting chemotherapy for women of childbearing potential, defined as those who have not been surgically sterilized or who have not been free of menses for at least 1 year. 9. Fertile male and female patients must be willing to use an effective contraceptive method before, during, and for at least 4 months after the huMNC2-CAR T cell infusion. Exclusion Criteria: 1. Patients requiring ongoing daily corticosteroid therapy at a dose of \>15 mg of prednisone per day (or equivalent). Pulsed corticosteroid use for disease control is acceptable. 2. Active autoimmune disease requiring immunosuppressive therapy is excluded unless discussed with the PI. 3. Major organ dysfunction defined as: 1. Serum creatinine \> 2 mg/dL 2. Bilirubin ≥ 1.5 mg/dL with the following exception: Patients with known Gilbert disease, serum bilirubin \> 3 mg/dL 3. AST or ALT ≥ 2.5 x upper institutional limit of normal with the following exception: Patients with known hepatic metastases, AST or ALT \> 3x upper institutional limit of normal 4. Patients with clinically significant pulmonary dysfunction, as determined by medical history and physical exam should undergo pulmonary function testing. Those with an FEV1 of \< 50 % of predicted or DLCO (corrected) \< 40% will be excluded. 5. Significant cardiovascular abnormalities as defined by any one of the following: i. NYHA class III or IV congestive heart failure, ii. clinically significant hypotension, iii. uncontrolled symptomatic coronary artery disease, or iv. a documented ejection fraction of \<45%. Any patient with an EF of 45-49% must receive clearance by a cardiologist to be eligible for the trial. 4. ANC \<1000/mm\^3. 5. Hemoglobin \<9 mg/dl (transfusion permitted to achieve this). 6. Platelet count \<75,000/mm\^3. 7. Treatment with investigational agent(s) within 30 days of planned lymphodepletion. 8. HIV seropositive. 9. Uncontrolled active infection. 10. Anticipated survival of \<3 months. 11. Breast-feeding women. 12. Patients who have a contraindication to cyclophosphamide chemotherapy. 13. Known second malignancy that is progressing or requires active treatment. 14. Untreated CNS metastases and/or carcinomatous meningitis. Patients with previously treated brain metastases may participate with documented stable disease as defined by no evidence of progression by imaging or symptoms for at least 4 weeks prior to enrollment. 15. Have psychiatric illness, social situation, or other medical condition that would preclude informed consent to limit compliance with study requirements, as determined by the investigator.