Make the Dough: In a food processor, process the flour, sugar and yeast for 2 seconds to combine. With the machine running, slowly add the water through the feed tube and process until dough is just combined and no dry flour remains, about 10 seconds. Let dough rest in the food processor for 10 minutes.
Add the oil and salt to the dough and process until the dough forms a satiny, sticky ball that clears the side of the workbowl, 30 to 60 seconds. Remove the dough from the bowl and knead briefly on a lightly oiled surface until smooth, about 1 minute. Shape the dough into a tight ball and place in a large, lightly oiled bowl. Cover tightly with plastic wrap and refrigerate for at least 24 hours (the dough can be refrigerated for up to 3 days).
Make the Sauce: Process all of the sauce ingredients in a food processor or blender until smooth, about 30 seconds. Transfer to an airtight container and refrigerate until ready to use.
Bake the Pizza: One hour before baking the pizza, adjust the oven rack to the second-highest position (the rack should be 4 to 5 inches below the broiler). Set a pizza stone on the rack and heat the oven to 500 degrees.
Remove the dough from the refrigerator and divide it in half. Shape each half into a smooth, tight ball. Place them on a lightly oiled baking sheet, making sure they are at least three inches apart. Cover them loosely with a piece of plastic wrap that has been sprayed with non-stick cooking spray. Set aside for 1 hour.
Coat one ball of dough with flour and place on a well-floured surface. Using your fingertips, gently flatten the dough into an 8-inch disk, leaving an inch or so of outer edge thicker than the center. Using your hands, gently stretch into a 12-inch round, working along the edges and giving the dough quarter turns as you stretch it. Transfer the dough to a well-floured pizza peel and stretch into a 13-inch round.
Spread ½ cup of the tomato sauce in a thin layer over the dough, then sprinkle with half of the Parmesan cheese and half of the mozzarella cheese. Top with pepperoni slices, if desired.
Slide the pizza carefully onto the stone and bake until the crust is well browned and the cheese is bubbly and beginning to brown, 10 to 12 minutes, rotating the pizza halfway through the baking time. Remove the pizza from the stone and place on a wire rack for 5 minutes before slicing and serving. Repeat steps to shape and bake second pizza.