Preheat the oven to 500° F. Arrange an oven rack in the top third of the oven and place a pizza stone on the rack. Let it heat for one hour. Remove the pizza dough from the refrigerator at least 30 minutes before baking.
Turn a large baking sheet upside down, or use a pizza peel, and dust the surface with flour. Form the dough into a 12-inch round.
Working quickly, spread one-fourth of the tomato sauce over the dough, then scatter one-fourth of the mozzarella over the pizza. Top with 5 to 7 basil leaves (depending on their size).
Jiggle the pizza gently on the pan to make sure it is not sticking (if it is, unstick it and sprinkle a little more flour under the area where it stuck). Carry the pan to the oven. Slide the pizza onto the stone; make sure to start at the stone’s back end so that the entire pizza will fit on the stone.
Cook the pizza for 3 minutes. Turn off the oven and turn on the broiler. Broil the pizza until golden, crisp, and a bit blistery and charred in places, 2 to 4 minutes (watch it carefully to see that it does not burn). Use tongs to slide the pizza to a large platter. Sprinkle with one-fourth of the Parmigiano-Reggiano and a pinch of sea salt. Before serving, drizzle with additional oil.
Repeat with the remaining dough balls and toppings.