Patients receive LV-NYESO TCR/sr39TK PBSC IV on day 0, and after approximately 24 hours, patients receive RV-NYESO TCR PBMC IV on day 1. Beginning on day 2, patients receive aldesleukin (interleukin-2 (IL) or IL-2) SC twice daily (BID) for up to 7 days. Patients undergo blood collection for safety and immune monitoring on days 0, 1, 3, 5, 7, 14, 30, 60, 90, and 120. Patients receive the PET tracer 18F-FHBG IV, and after 1 hour, undergo PET/computed tomography (CT) on days 25 and 120.