Prepare Dough: Combine flour, sugar, and yeast in a food processor. Pulse briefly to mix. While running the processor, slowly pour in ice water and blend until a shaggy dough forms, around 10 seconds. Let rest for 10 minutes.
Knead Dough: Add vegetable oil and salt. Blend until the dough becomes smooth and elastic, about 30 seconds. Transfer to a lightly oiled surface and knead for 1 minute. Shape into a ball, place in a greased bowl, cover tightly with plastic wrap, and refrigerate for 24 hours (or up to 3 days).
Make Sauce: Blend all sauce ingredients in a food processor for roughly 15 seconds, scraping the sides as needed. Transfer to a container and refrigerate until needed.
Preheat Oven: About an hour before baking, position a rack 4½ inches below the top heating element and place your baking stone or sheet on it. Preheat oven to 500°F. Meanwhile, let the dough rest at room temperature on a greased baking sheet, covered, for 1 hour.
Shape Dough: Lightly flour your counter and hands. Flatten the dough into an 8-inch round, keeping the edge slightly thicker. Stretch gradually into a 12-inch circle, turning as you go to maintain the shape.
Assemble Pizza: Move dough to a floured peel or surface. Stretch slightly more to form a 13-inch round. Spread sauce evenly, leaving a ¼-inch border. Sprinkle with Parmesan, then mozzarella.
Bake: Slide pizza onto the preheated stone and bake for 10–12 minutes, rotating halfway through, until the crust is golden and cheese is bubbling. Cool on a wire rack for 5 minutes before slicing.