Position racks in the upper and lower thirds of the oven and preheat to 475 degrees . On a parchment-lined rimmed baking sheet, toss the butternut squash with 1 tablespoon olive oil; season with salt and pepper. Roast in the upper third of the oven until golden and slightly softened, about 12 minutes.
Meanwhile, in a small bowl, combine the garlic and parsley with 2 tablespoons olive oil; season with salt and pepper. In a large bowl, toss the radicchio and onion with the remaining 1 tablespoon olive oil and season with salt and pepper; add the roasted squash and toss, reserving the baking sheet and discarding the parchment.
Line the reserved baking sheet with a clean sheet of parchment. Stretch the dough to fit the baking sheet and smear the garlic mixture on top, leaving a 1/2-inch border. Sprinkle the cheese on top, followed by the squash mixture. Bake in the lower third of the oven until the crust is deep golden, about 15 minutes.