Leon Elnekave, the affable leader of the dwindling Jewish community of Bursa, Turkey’s old Ottoman capital, met my wife and me on a narrow, cobbled lane and unlocked the black wrought iron gate to the 500-year-old Gerush Synagogue. As he led us into the landscaped courtyard, my wife, Sallie, suddenly began to cry. “I don’t know what it is,” she said, “but I feel like I’ve been here before.” An odd reaction, I thought to myself, for an American woman who converted to Judaism from Presbyterianism in middle age. But immersing yourself in a mystical place like this can provoke unexpected emotions.

My own motivation for the visit was more pedestrian than spiritual: I had a suburban, white-bread, Ashkenazi curiosity about our Sephardic cousins and their exotic history....