Set a pizza stone in the oven and preheat the oven to 500°, allowing at least 45 minutes for the stone to heat.
Brush both sides of one tortilla with olive oil. Place the tortilla on a lightly floured pizza peel and sprinkle with 1 tablespoon of the Parmigiano-Reggiano cheese. Thinly slice 1 tomato; arrange on the tortilla. Season with salt and pepper and top with 1 cup of the mozzarella. Drizzle lightly with olive oil.
Slide the tortilla onto the hot pizza stone and bake for about 8 minutes, until the pizza is crisp on the bottom and the cheese is bubbling. Garnish with 1/4 cup of the shredded basil, cut the pizza into 8 wedges and serve. Repeat with the remaining tortillas, tomatoes, cheeses and basil.