Pasta – Red and blue Antimo Caputo 00 flour

In my experience the primary difference is that Calzones have Ricotta (and possibly Mozzarella) and Strombolis only have Mozzarella. In the Philadelphia area, both are folded over pizzas, basically.

Growing up, my mother made Stromboli and it was rolled, not folded. I rarely see that in a pizza shop these days, but I do prefer that in a Stromboli.

Edit: Just read systempuntoout's link to Wikipedia and it confirmed something that I was going to post and then deleted. When I was growing up, Strombolis were made from bread dough and Calzones were made from pizza dough; nowadays all of the pizza shops just use pizza dough. I remember bread stores as being the primary place to get Strombolis growing up because they were the ones that made the dough!

I recommend starting with a chilled dough, puncturing with a knife or fork (see below), and cooking in a greased pan initially then bricking.

Cooking directly on the stone the entire time works better for thinner crusts than mid-dough. By mid-dough, I am referring to the ones that plump up a bit and finish between 1/4" - 1/2" in thickness. They can have both uniform, almost cake-y cross-sections, or air pockets and bubbles depending on how kneaded they are (more kneading means cake-ier, denser dough).

I worked in a shop that made a mid-dough similar to what you describe. It was a standard dough recipe (flour, water, yeast), but I have had success replicating it at home at 450'F with beer and less kneading. At home I use a pan and then finish either on the rack or a stone. In any case, you get a dough that tears nicely and has a good chew.

The dough itself was mixed in the morning and kept refrigerated for as long as 12 hours as individual shells in greased pizza pans covered with saran wrap. To prepare a pizza, we pulled the shell, punched it with a fork (puncture a ring around the edge to create a crust, puncture the inside to allow for air). Dress with sauce and toppings.

We had a stone-bottomed oven, but initially the pizza is baked in the oven in the pan until the bottom of the dough has hardened enough to get "bricked" (the dough should be rigid enough as to be removed from the pan with one spatula). At this point you remove the dough from the pan with a spatula and place it directly on the brick. Cook until the top of the dough browns, and the cheese and other toppings are evenly browned.