This is an open-label phase 1 study to assess the safety and feasibility of autologous T cells expressing a single-chain scFv targeting GFRα4 with tandem TCR/CD3ζ and 4-1BB (TCRζ/4-1BB) co-stimulatory domains (referred to as "CART-GFRa4 cells") in patients with incurable medullary thyroid cancer (MTC).

Inclusion Criteria:

1. Signed, written informed consent

2. Male or female age ≥ 18 years

3. Histologically or cytologically confirmed diagnosis of medullary thyroid cancer (MTC).

4. Incurable recurrent/metastatic disease that is progressive after at least 1 prior tyrosine kinase inhibitor (TKI) containing regimen, or the patient was intolerant of or declined such therapy.

5. Adequate organ function defined as:

   1. Serum creatinine ≤ 2.5 mg/dl or estimated creatinine clearance ≥ 30 ml/min and not on dialysis.
   2. AST ≤ 5x upper limit of normal range and total bilirubin ≤ 2.0 mg/dl; except for patients in whom hyperbilirubinemia is attributed to Gilbert's syndrome.
   3. Left Ventricular Ejection Fraction (LVEF) ≥ 45% confirmed by ECHO/MUGA
   4. Must have a minimum level of pulmonary reserve defined as ≤ Grade 1 dyspnea and pulse oxygen greater than 92% on room air.

6. ECOG Performance Status that is either 0 or 1.

7. Toxicities from prior therapies must have recovered to grade ≤ 2 according to the CTCAE 5.0 criteria or to the patient's prior baseline.

8. Patients must have evaluable disease as defined by RECIST 1.1.

9. Subjects of reproductive potential must agree to use acceptable birth control methods.

Exclusion Criteria:

1. Evidence of active hepatitis B or hepatitis C infection. The following would not qualify as an active infection, thus would not exclude the subject from participating

   1. Positive HBV serology with undetectable viral load and ongoing antiviral prophylaxis for potential HBV reactivation.
   2. Positive HCV serology with quantitative PCR for plasma HCV RNA below the lower limit of detection, with or without concurrent antiviral HCV treatment.

2. Any other active, uncontrolled infection.

3. Any prior history of moderate to severe (Grade 2 or higher) pneumonitis.

4. Subjects with chronic kidney disease with Grade 2 or higher renal impairment (eGFR or CrCl 59-30 ml/min/1.73 m2).

5. Class III/IV cardiovascular disability according to the New York Heart Association Classification.

6. Clinically apparent arrhythmia or arrhythmias that are not stable on medical management within two weeks of physician-investigator confirmation of eligibility.

7. Planned concurrent treatment with systemic high dose corticosteroids. Patients may be on a stable low dose of steroids (≤10mg equivalent of prednisone). Use of inhaled steroids is allowable. Corticosteroid treatment as anti-emetic prophylaxis on the day of lymphodepleting chemotherapy administration is allowed per institutional practice.

8. Any moderate to severe skin rash or allergies requiring systemic treatment.

9. Receipt of immune checkpoint inhibitors within 2 months prior to physician-investigator confirmation of eligibility - Retired with Protocol Version 3.

10. Pregnant or nursing (lactating) women.

11. Active autoimmune disease requiring systemic immunosuppressive treatment equivalent to ≥ 10mg daily of prednisone. Patients with autoimmune neurological diseases (such as MS or Parkinson's) will be excluded.

12. Have any history of prior or active central nervous system (CNS) involvement (e.g., leptomeningeal disease, parenchymal masses) with MTC. Screening for this (e.g., with lumbar puncture and/or brain MRI) is not required unless suspicious symptoms and/or radiographic findings are present. Subjects with calvarial metastatic disease that extends intracranially and involves the dura will be excluded, even if CSF is negative for MTC.

13. Known seizure disorder or history of prior seizures requiring medication.

14. History of allergy or hypersensitivity to study product excipients (human serum albumin, DMSO, and Dextran 40).