EARLIER this year, when asked what Italian cuisine he’d like to see better represented in New York, Mario Batali said, without hesitation, “the food of Apulia.” The rustic cooking of the region in the heel of Italy takes a distant back seat to the food of noble Tuscany and lusty Naples, and is represented on few menus in New York. Nicola Marzovilla, who is from Polignano a Mare, a coastal town in Apulia, and owns I Trulli in Manhattan, attributes that to the region’s cuisine being “the ultimate home cooking,” he said. “It’s not easy to do the food in...