In a medium bowl, whisk together the all-purpose flour, sugar, yeast, and salt. Add the warm water and olive oil, stirring the mixture with a wooden spoon until the dough begins to come together. The dough will appear shaggy and dry, but don't worry.

Scrape the dough onto a well-floured countertop and knead for about three minutes. The dough will quickly come together and become sticky. Dust with flour as necessary (you may need to do this 2-3 times, depending on humidity levels) so that the dough becomes slightly tacky but not sticky to the counter. After 3 minutes, the dough should be smooth, slightly elastic, and tacky. Lightly grease a large mixing bowl with olive oil and place the dough inside.

Cover the bowl with a kitchen towel or plastic wrap and let the dough rise in a warm, dry place for 2 hours, or until it doubles in size. Proofing Tip: If your kitchen is cold, heat a large heatproof measuring cup of water in the microwave for 2-3 minutes. This will create a warm environment. Remove the cup, then place the bowl with the dough inside the microwave until it rises. [If preparing the dough in advance, check the note section for freezing instructions.]

Place the pizza steel (or stone) on the second-to-top rack of your oven (about 8 inches from the broiler) and preheat to 550°F (285°C) for at least 1 hour. If your oven can't reach 550°F (285°C) or if you're using a delicate pizza stone, preheat to a maximum of 500°F (260°C).

While the oven is preheating, assemble your ingredients. In a small bowl, combine the pureed tomatoes, minced garlic, extra virgin olive oil, pepper, and salt. Set aside. In another small bowl, prepare the cubed mozzarella (pat it dry with a paper towel to remove excess moisture). Keep the basil leaves and grated Parmigiano-Reggiano cheese nearby for easy access.

Divide the dough into two equal portions. It will deflate slightly, but that's okay. Place the dough on a floured plate or countertop, cover it with plastic wrap, and let it rest for 5-10 minutes.

Sprinkle a pizza peel (or the back of a half sheet pan, though it’s trickier) with semolina and a dusting of all-purpose flour. Gently stretch one ball of dough into a roughly 10-inch circle using both hands (don’t worry if it’s not perfectly round). If the dough springs back or feels too elastic, let it rest for another 5 minutes. The edges should be slightly thicker, and the center should be thin (you should be able to see light through it when held up). Carefully transfer the dough onto the semolina- and flour-dusted pizza peel or baking sheet.

Lightly drizzle or brush olive oil on the dough with your fingertips (about 1 teaspoon). Using a large spoon, spread about ½ cup of tomato sauce onto the pizza dough, leaving a ½ to ¾-inch border. Sprinkle with a tablespoon of Parmigiano-Reggiano. Add half of the cubed mozzarella, evenly distributing it over the pizza. Tear a few basil leaves by hand and scatter them over the pizza. If needed, stretch the dough’s edges to make it thinner.

Gently slide the pizza from the peel onto the preheated baking stone. Bake for 7-8 minutes, or until the crust is golden, the cheese is bubbling and caramelized, and the pizza edges are golden brown. Note: If you want more color, finish the pizza under low or medium broil, but watch carefully!

Remove the pizza from the oven using the pizza peel, and transfer it to a wooden cutting board or foil. Drizzle with olive oil, sprinkle with grated Parmigiano-Reggiano, and top with a chiffonade of fresh basil. Slice and serve immediately. Repeat the process for the second pizza.

If you’re serving two pizzas, place the first pizza on a separate baking sheet while preparing the second. In the final minutes of cooking, place the first pizza in the oven (on a rack below the pizza stone) to keep it hot for serving. Alternatively, serve the first pizza fresh out of the oven, then prepare the second pizza once the first is finished. Either way, the pizza is best enjoyed right after baking!