1 Prepare and heat the oven: If you have one, place a pizza stone on a lower rack in your oven. Preheat the oven to 550F or as hot as it will go.
2 Pre-shape the pizza rounds and let them rest: Turn out the dough onto a generously floured counter and cut it into 4 pieces. Working with 1 piece at a time, shape into a ball and then press flat into a 1-inch thick disk, dimpling with your fingertips as you flatten it. This isn’t the final shaping; just a pre-shaping to help make the dough easier to work with later on.
Cover the first disk with a clean dishtowel, and repeat with the remaining pieces of dough. Let the disks of dough rest for 10 to 15 minutes while you prepare the vegetables. This gives the dough time to relax before shaping.
3 Precook the potatoes: Mound the potato slices in the middle of a baking sheet. Sprinkle with 1 tablespoon of oil, and salt and pepper to taste. Toss together and spread in an even layer with no pieces overlapping.
Bake for 10 minutes, or until the potatoes are tender. Let cool briefly.
4 Prepare the asparagus: Trim about 1/2 inch from the bottom of the asparagus spears. With a vegetable peeler, shave the spears into long, thin strips: hold a spear by the bud end and scrape down the length with the peeler. Turn the stalk often until you can no longer scrape it. Cut the remaining core of the spear into 1-inch pieces.
In a bowl, toss all of the asparagus with the remaining 2 tablespoons of oil, salt and pepper. Set aside.
5 Shape the pizza rounds: Sprinkle a layer of cornmeal over a pizza peel, a rimless baking sheet, or the back of a baking sheet. (If you don’t have a pizza stone in your oven, just bake the pizza on the baking sheet.)
Hold up one piece of dough at the edge and turn it as if you were holding a steering wheel, letting gravity pull the dough down (alternatively, roll with a rolling pin). Keep stretching it until the round is roughly a 12-by-8-inch oval or a 10-inch circle.
Place it on the pizza peel and shuffle the pan back and forth to ensure the dough is not sticking (sprinkle extra cornmeal underneath if it sticks.)
Leave the remaining rounds covered.
6 Add the toppings and bake the pizza: Arrange one-fourth of the potatoes, one fourth of the cheese, and one fourth of the asparagus on top of the pizza round.
Slide the pizza onto the baking stone to bake (or bake on a baking sheet if you don’t have a pizza stone).
Bake for 10 to 12 minutes, or until the crust is golden and the cheese is bubbling. Use a pizza peel to remove the pizza from the oven and transfer it to a cooling rack.
Prepare and bake the other pizzas in the same manner. If your pizza stone is large enough, you can bake two pizzas side-by-side at the same time.
7 To serve: Pizzas are best served hot from the oven! Rather than waiting for all the pizzas to be done, slice and serve each pizza as it comes out of the oven. If you’d rather serve pizzas all at once, briefly re-warm any pizzas that cooled by placing them back in the oven for a minute or two.