Step 1

Position 1 rack in top third and 1 rack in bottom third of oven and preheat to 450°F. Lightly flour 2 baking sheets. Place dough on work surface; let stand until room temperature, about 20 minutes. Divide dough in half. Press and stretch each piece out on lightly floured surface to 5-inch round. Sprinkle each with 1/3 cup Piave cheese, 3/4 teaspoon rosemary, and 1/4 teaspoon crushed red pepper; sprinkle with coarse salt. Roll each piece of dough out to 10-inch round, pressing in seasonings. Transfer dough rounds to prepared baking sheets.

Step 2

Heat 1 tablespoon oil in large nonstick skillet over medium-high heat. Add sausage. Sauté until brown, breaking into 1/2-inch pieces with back of spoon, about 5 minutes. Using slotted spoon, transfer sausage to bowl. Add onion to skillet. Sauté until crisp-tender, about 2 minutes; transfer to plate. Add remaining 1/2 tablespoon oil to skillet. Add mushrooms and remaining 3/4 teaspoon rosemary; sprinkle with salt and pepper. Sauté until brown, about 5 minutes. Leaving 1/2-inch plain border, top each dough round with 3/4 cup mozzarella, then onion, sausage, and mushrooms.

Step 3

Bake until crust bottoms are crisp and brown, reversing sheets after 10 minutes, about 20 minutes total. Using large spatula, transfer pizzas to work surface. Sprinkle each with 2 tablespoons mozzarella, then parsley, if desired.