When Gershon Pincus turned 60, he decided he wanted to give something back to his country. The Brooklyn-born father of four had maintained a successful private dental practice in New York City for 35 years. As he would later attest in an affidavit, “I can think of no better way to experience the sunset of my career than by using my professional skills as a dentist to assist those who have chosen to serve in the United States military.”

So Dr. Pincus turned to the USAJobs website, found an opening for part-time work at an off-base...