Prepare the Pizza Dough.

Preheat the oven to 350 degrees.

Cut the butternut squash in half lengthwise and brush the cut sides with olive oil. Place the squash cut side down on a parchment-lined baking sheet and bake for 1 hour and 30 minutes. Remove from the oven. When cool enough to handle, peel the skin off the squash. Slice lengthwise into ½-inch slices. Reserve.

Increase the oven temperature to 425 degrees. Heat your grill and grill rack. Place one 12-by-9-inch pizza dough round on a floured pizza paddle or on a floured inverted baking sheet.

When your grill is hot, gently slide the dough onto the grill and brush the top with olive oil. Within a minute the dough will begin to puff .Grill for 1 to 2 minutes more.

With tongs, lift the edge of the dough to check the color underneath. When the underside of the dough becomes golden brown and you see grill marks, flip the dough over. Grill for 1 minute longer and remove the grilled dough with the pizza paddle. Repeat with the other pizza dough round.

Brush the first grilled side with olive oil and spread half the fontina evenly on top. Then lay half of the squash slices over the cheese and sprinkle with half of the rosemary and Parmigiano.

Drizzle with olive oil and slide the pizza back onto the grill. Grill for 2 minutes longer.

Season with salt and freshly ground dried chiles to taste. Slice and serve immediately. Repeat with the other pizza.