My plan is to get a pizza steel and do the cooking in my Big Green Egg. I can get the egg up to at least 700 F and I should be able to cook the pizza in a few minutes at that temp. The kamado ceramic body of the BGE will retain heat well and cook from both top and bottom. The oak wood should add nice flavor, too.
I baked three pies in my BGE last night. Delicious. Pizza stone on grate on plate setter and bake 2.5 minutes at 650-700, rotate 180 degrees, and another 2-3 minutes. Delicious. Easy shortcut is to use Trader Joe's crust.