Deep in the Tuscan countryside, between a hill and a bleating sheep, Gianluca Tonelli tended to 17 pounds of pastrami soaking in a plastic container filled with brine. With his trademark porkpie hat and gray goatee, Mr. Tonelli, a worshiper of what he called “pastrami culture” — “We started a klezmer band!” — sidestepped Dante, his family’s pet pig, and loaded freshly chopped cherry wood chips into a smoker. In the house, he showed off heavy-duty cooking devices, including a pressure cooker in which he aromatizes the pastrami with cloves and Tuscan red wine, before selling it from a bright...