What should I order that isn't on the American Italian menu?

Order what the region makes, and ignore everything you know as "Italian food" at home. In Rome: cacio e pepe, carbonara, amatriciana, carciofi alla giudia, supplì. Florence and Tuscany: ribollita, pappa al pomodoro, pici, bistecca to share. Venice: cicchetti at a bacaro, sarde in saor, risotto. Naples and the south: pizza, ragù that cooked for six hours, sfogliatella. Bologna: tagliatelle al ragù — never spaghetti. Sicily: arancine, pasta con le sarde, cannoli filled at the moment you order. If a menu offers all of those at once, you're in the wrong place.