I like this idea. I remember having lunch on a vine-shaded terrace in Frascati with views as far as Rome in the distance.... It was a hotel though, not a restaurant.   Another possibility in that general direction but a bit further might be to visit Tivoli. You could visit either Villa d'Este or Hadrian's villa and there is a wonderful place for lunch at the top of the hill in town with wonderful terrace that overlooks a couple of ancient Roman temples. Great food too. (It's a hotel al