Our preferred brand of whole-milk mozzarella is Dragone. You can shape the second dough ball while the first pizza bakes, but don’t top the pizza until right before you bake it. If you don’t have a baking stone, bake the pizzas on an overturned and preheated rimmed baking sheet. It is important to use ice water in the dough to prevent overheating the dough while in the food processor. Semolina flour is ideal for dusting the peel; use it in place of bread flour if you have it.
For the Dough: In a 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, 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.