Let’s see if I got the gist of this community right.

(And yes I did eat at All’antico Vinaio and wanted it again.)

The bread they use is schiacciata. It’s like focaccia but not - different hydration and made with something called a “poolish”, where you take a certain amount of yeast/flour/water and let it ferment for an extended period of time (multiple hours/overnight) then add more water/flour and let it it proof again. Because the hydration is so high, it is sticky as hell, so be careful when working with it.

The cheeses, obviously, are not just regular sliced cheese. What they frequently use is stracciatella, which is mozzarella mixed with cream and cooked. It’s the center for burrata.
But there’s also a “cheese cream” like “pecorino cream” which seems to be a mornay sauce (aka grated/shredded cheese + bechamel (roux + milk)). Looks to be a way to make sure cheese doesn’t go to waste, makes it go further, and is spreadable. (And hecking delicious)

Finally there are other creams like pistachio cream, which is finely chopped pistachios mixed with Parmesan or Pecorino Romano and Olive oil (and maybe some lemon). It’s delicious and entirely different from what I as an American use as sandwich condiments.