In a medium bowl, add 2 1/2 cups of the all-purpose flour, the granulated sugar, the yeast, and 3/4 teaspoon of the salt and whisk to combine.
Add the warm water and 1 tablespoon of the olive oil to the flour mixture and stir with a wooden spoon until the dough just begins to combine and is shaggy and dry.
Use the extra all-purpose flour to generously dust a work surface.
Place the dough on the work surface and knead, sprinkling it as needed with the extra all-purpose flour, until it becomes smooth, slightly elastic, and tacky but not sticking to the surface, about 3 minutes.
Lightly grease a large bowl with the extra olive oil and add the kneaded dough.
In a warm, dry area, cover the kneaded dough with a clean kitchen towel and let it rise until it has doubled in size, about 2 hours.
On the middle rack of the oven, about 8-inches from the broiler element, add a pizza steel or pizza stone.
Preheat the oven to 550 degrees F, about 1 hour. If the oven does not reach 550 degrees F or if using a delicate pizza stone, heat to a maximum of 500 degrees F.
While the oven and the pizza steel are preheating, in another medium bowl, add the tomatoes, the garlic, 1 teaspoon of the olive oil, the salt, and the black pepper and stir to combine. Set it aside.
On a large plate, separate the dough into 2 equal-sized portions and gently cover each of them with plastic wrap. Let them rest for about 5-10 minutes.
Lightly sprinkle a pizza peel or the back of a half sheet pan with the semolina flour and the extra all-purpose flour.
Use your hands to gently stretch 1 of the pizza dough balls into a 10-inch circle; it does not need to be perfectly uniform. If the dough springs back or is too elastic, let it rest, about 5 minutes. The center of the dough should be thin enough to see light through and the edges should be slightly thicker.
Gently transfer the dough to the prepared pizza peel.
Lightly brush 1 teaspoon of the olive oil over the dough.
Use a large spoon to evenly and thinly spread about 1/2 cup of the tomato sauce mixture over the pizza dough, leaving a 1/2-3/4-inch border on all sides of the dough without the sauce.
Evenly sprinkle 1 tablespoon of the Parmigiano-Reggiano cheese over the tomato sauce layer.
Evenly add 1/2 of the mozzarella cubes on top of the Parmigiano-Reggiano cheese layer, then top with 1/2 of the torn basil leaves.
Use your hands to stretch the sides of the pizza dough to make it thinner.
Gently slide the prepared pizza onto the heated baking stone and bake until the crust is golden, the cheese is bubbling and caramelized, and the edges of are golden-brown, about 7-8 minutes.
Use the pizza peel to carefully transfer the pizza from the oven to a cutting board.
Drizzle the top of the pizza with 1 teaspoon of the olive oil, the extra Parmigiano-Reggiano cheese, 1 of the shredded basil leaves, and the red pepper flakes.
Prepare and bake the second pizza using the remaining ingredients.
Slice and serve.