Haji Din Mohammed met with the Taliban for the first time on the public record on July 7, 2015, in the town of Murree, Pakistan, just outside Islamabad. It was Ramadan, the Muslim month of fasting. After sunset, he and his colleagues — delegates from the High Peace Council, the Afghan government’s official negotiating body — sat down for a customary iftar dinner at their hotel before heading over to a nearby golf resort.

Din Mohammed, who is in his early sixties, has a white beard and rheumy eyes. That evening, he wore a lungi, a turban that indicated his elite social…