In food processor fitted with metal blade, process flour, sugar, and yeast until combined, about 2 seconds.
With machine running, slowly add water through feed tube; process until dough is just combined and no dry flour remains, about 10 seconds.
Let dough stand 10 minutes.
Add oil and salt to dough and process until dough forms satiny, sticky ball that clears sides of work bowl, 30 to 60 seconds.
Remove dough from bowl and knead briefly on lightly oiled countertop until smooth, about 1 minute.
Shape dough into tight ball and place in large, lightly oiled bowl. Cover tightly with plastic wrap and refrigerate for at least 24 hours and up to 3 days.
One hour before baking pizza, adjust oven rack to second highest position (rack should be about 4 to 5 inches below broiler), set pizza stone on rack, and heat oven to 500 degrees.
Remove dough from refrigerator and divide in half. Shape each half into smooth, tight ball. Place on lightly oiled baking sheet, spacing them at least 3 inches apart; cover loosely with plastic wrap coated with nonstick cooking spray; let stand for 1 hour.
Coat 1 ball of dough generously with flour and place on well-floured countertop. Using fingertips, gently flatten into 8-inch disk, leaving 1 inch of outer edge slightly thicker than center.
Using hands, gently stretch disk into 12-inch round, working along edges and giving disk quarter turns as you stretch. Transfer dough to well-floured peel and stretch into 13-inch round.
Using back of spoon or ladle, spread 1/2 cup tomato sauce in thin layer over surface of dough, leaving 1/4-inch border around edge.
Sprinkle 1/4 cup Parmesan evenly over sauce, followed by 1 cup mozzarella. Too much cheese will create a soggy pizza.
Slide pizza carefully onto stone and bake until crust is well browned and cheese is bubbly and beginning to brown, 10 to 12 minutes, rotating pizza halfway through (if you can).
Remove pizza and place on wire rack for 5 minutes before slicing and serving.
TOPPING TIPS: We like our Thin-Crust Pizza simply dressed with tomato sauce and handfuls of shredded mozzarella and Parmesan, but additional toppings are always an option--provided they're prepared correctly and added judiciously. (An overloaded pie will bake up soggy.) Here are a few guidelines for how to handle different types of toppings:
HEARTY VEGETABLES Aim for a maximum of 6 ounces per pie, spread out in a single layer. Vegetables such as onions, peppers, and mushrooms should be thinly sliced and lightly sautéed (or microwaved for a minute or two along with a little olive oil) before using.
DELICATE VEGETABLES AND HERBS Leafy greens and herbs like spinach and basil are best placed beneath the cheese to protect them or added raw to the fully cooked pizza.
MEATS Proteins (no more than 4 ounces per pie) should be precooked and drained to remove excess fat. We like to poach meats like sausage (broken up into 1/2-inch chunks), pepperoni, or ground beef for 4 to 5 minutes in a wide skillet along with 1/4 cup of water, which helps to render the fat while keeping the meat moist.
NOTE as with all breads, the ratio of wet to dry is critical and more accurately measured by weight).