How to Make It
Set a pizza stone on a rack in the top third of the oven. Preheat the oven to 500° for at least 45 minutes. Meanwhile, remove the dough ball from the refrigerator and let stand for 20 minutes.
Working on a floured surface and using your fingers, press and stretch the dough out to a 10-inch round, working from the center toward the edge; avoid pressing on the outermost edge. Transfer the dough to a lightly floured pizza peel. Spread the Parmigiano-Reggiano and Fontina cheeses onto the dough, making sure to leave a 1-inch border around the edge, and top with the squash, rosemary and chiles.
Turn the oven to broil for 5 minutes, then return it to 500°. Slide the pizza onto the pizza stone, opening and closing the oven door as quickly as possible. Bake until the bottom is lightly charred and the toppings are bubbling, about 6 minutes for a chewier crust and 8 minutes for a crispier one; avoid opening the oven door during baking. Top the pizza with the arugula before serving.