The internet is a wonderful thing. You can type: "gluten free pizza Chicago" into the googler and find places like: Marcello's Father & Son Restaurant offering both deep dish and thin crust gluten free offerings. You can even order online.
Do you think I could resist?
Deep dish with mushrooms, onions & sausage:
Thin crust with spinach, mushrooms, & artichokes:
Flavor-wise, the thinner crust won out. I am pretty sure its the same dough, just shaped differently, but the thicker crust of the deep dish tasted a bit odd or off - though both held up very well, even with delivery they didn't get soggy or fall apart.
The left overs were even better.