Preheat your oven to the hottest temperature it goes, or about 500 in most cases. If you use a pizza stone, have it in there.
Prepare asparagus: No need to snap off ends; they can be your 'handles' as you peel the asparagus. Holding a single asparagus spear by its tough end, lay it flat on a cutting board and using a vegetable peeler, create long shavings of asparagus by drawing the peeler from the base to the top of the stalk. Repeat with remaining stalks and discard tough ends. Toss peelings with olive oil, salt and pepper in a bowl.
Assemble and bake pizza: Roll or stretch out your pizza dough to a 12-inch round. Transfer to a floured or cornmeal-dusted pizza peel or tray. Sprinkle pizza dough with Parmesan, then mozzarella. Pile asparagus on top. Bake pizza for 10 to 15 minutes, or until edges are browned, the cheese is bubbly and the asparagus might be lightly charred. Remove from the oven and immediately sprinkle with scallions, then slice and eat.
