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1. Place a baking stone, if using, on the middle shelf of the oven (unless you know your oven well enough to place it on a different shelf) and preheat to 400°F (200°C). If you do not have a baking stone, position 1 oven shelf in the center of the oven and a second in the lowest position.
2. Oil two 10-inch round cake pans. Roll out each dough ball with a rolling pin, working from the center outward, into a disk about 14 inches in diameter. If the dough resists rolling or becomes too springy and elastic, cover it with a towel and let it rest for 5 minutes before resuming. Lay a finished disk over a prepared pan, lower it gently into the pan, and then press the dough snuggly into the bottom, corners, and then up the sides. Ideally the dough will drape slightly over the rim; crimp the overhang to form a thick edge around the top of the rim. Prick the entire surface with a fork to minimize bubbling in the oven.
3. If you like a thin crust, bake immediately. If you prefer a thicker crust, cover the dough-lined pans with a kitchen towel and let sit at room temperature for 30 to 60 minutes. Bake the crusts on the middle shelf of the oven for 3 to 4 minutes to set them. The rim of the dough will slide back down into the pan somewhat, but this is normal and will form the edge.
4. Place each crust in its pan on the counter and let cool for at least 5 minutes. Evenly spread the mozzarella cheese in the bottom, then ladle in the tomato sauce to fill the pans. Sprinkle the grated aged cheese evenly over the top of the sauce.
5. Place the pans in the oven on the baking stone or, if not using a stone, on the lowest oven shelf. Immediately reduce the heat to 375°F (190°C) and bake for 15 minutes. Then, if baking on a stone, rotate the pan 180 degrees. If baking directly on the shelf, move the pans to the middle shelf and rotate them 180 degrees. In either case, continue to bake for 20 to 25 minutes longer, or until the edges of the crust are a deep brown (but not burned) and the cheese is golden.
6. Remove the pizzas from the oven and place on a cooling rack for 5 minutes. Remove the pizzas from the pans by loosening the crust with a metal spatula and then gently sliding the spatula under the pizza to lift it out. Serve the pizzas on a cutting board, or cut and serve directly from the pan.
RICOTTA AND CHEESE: Cut the quantity of mozzarella cheese in the master recipe to 1 1/2 cups and stir it together with 1 1/2 cups ricotta cheese or full-fat or low-fat small-curd cottage cheese. Add 2 eggs, beaten, and 1 teaspoon salt and stir until all the ingredients are evenly distributed. Fill the prebaked shells with the cheese mixture and ladle the tomato sauce over the cheese to fill the shell. Omit the grated aged cheese, if desired. Bake as directed.
ITALIAN SAUSAGE: This is the most popular deep-dish topping in restaurants. You can make it two different ways. For my favorite method, shape 2/3 pound Italian pork sausage meat (removed from the casing) into 2 patties the diameter of the deep-dish pans. Layer the mozzarella cheese, the tomato sauce, and the dry aged cheese in the pan as directed, then lay the patty over the top like a lid. Brush the patty with olive oil and bake as directed. The sausage will have taken on a dark, rich color when the pizza is ready.
For the second method, which is less fatty, fry 2/3 pound Italian sausage meat in a frying pan over medium heat, stirring to break up any lumps, for about 5 minutes, or until lightly browned. Transfer the contents of the pan to a sieve placed over a bowl to drain off the fat. Layer the mozzarella cheese and the tomato sauce in the pan, scatter the cooked sausage over the top, and then finish with the grated aged cheese. Bake as directed.
You can substitute ground beef, lamb, chicken, or turkey seasoned with salt and pepper (about 1/2 teaspoon salt and 1/8 teaspoon coarsely ground black pepper) and minced fresh garlic or granulated garlic powder to taste for the sausage in either method.
SPINACH: In a bowl, stir together 12 ounces thawed, frozen chopped spinach, squeezed dry, with 2 1/2 cups shredded full-fat or low-fat low-moisture mozzarella cheese. Season to taste with minced fresh garlic or granulated garlic powder to taste, if desired. Substitute this mixture for the mozzarella cheese in the master recipe and then continue as directed.