Pizza Dough: In a medium bowl, whisk together the all-purpose flour, sugar, yeast and salt. Add the warm water and olive oil, and stir until the dough just begins to come together.
Scrape the dough onto a well-floured counter top and knead the dough for three minutes. It should quickly come together and begin to get sticky. Dust the dough with flour as needed – it should be slightly tacky, but should not be sticking to your counter top. After three minutes, the dough should be smooth, slightly elastic, and tacky. Lightly grease a large mixing bowl with olive oil, and place the dough into the bowl.
Cover the bowl with a kitchen towel (or plastic wrap) and allow the dough to rise in a warm, dry area for 2 hours or until the dough has doubled in size. If your kitchen is very cold, one great tip that I do all the time is to heat a large heatproof measuring cup of water in the microwave for 2-3 minutes. This creates a nice warm environment and I’ll immediately remove the cup and place the bowl with the dough in the microwave until it has risen.
Preheat Oven and Pizza Stone: Place a pizza stone on the center (or top third) rack of your oven, and preheat the oven and pizza stone to 550 degrees Fahrenheit (for at least 30-45 minutes). If your oven does not go up to 550 degrees, heat it to the absolute maximum temperature that it can go.
As the oven is preheating, assemble the ingredients. In a small bowl, stir together the pureed tomatoes, minced garlic, extra virgin olive oil, pepper, and salt. Set aside another small bowl with the cubed mozzarella cheese (pat the cheese with a paper towel to remove any excess moisture). Set aside the basil leaves and grated parmigiano-reggiano cheese for easy grabbing.
Separate the dough into two equal-sized portions. It will deflate slightly, but that is OK. Place the dough on a large plate or floured counter top, cover gently with plastic wrap, and allow the dough to rest for 5 to 10 minutes.
Sprinkle a pizza peel (or baking sheet) with a tablespoon of cornmeal. Gently stretch one ball of pizza dough into roughly a 10-inch circle. If the dough springs back or is too elastic, allow it to rest for an additional five minutes. The edges of the dough can be slightly thicker, but make sure the center of the dough is thin (you should be able to see some light through it if you held it up). Gently transfer the dough onto the cornmeal-dusted pizza peel or baking sheet.
Drizzle or brush the dough lightly with olive oil. Using a large spoon, add roughly 1/2 cup of the tomato sauce onto the pizza dough, leaving a 1/2-inch or 3/4-inch border on all sides. Use the back of the spoon to spread it evenly and thinly. Sprinkle a tablespoon of parmigiano-reggiano cheese onto the pizza sauce. Add half of the cubed mozzarella, distributing it evenly over the entire pizza.
Using your hands, tear a few large basil leaves, and sprinkle the basil over the pizza. At this point, I’ll occasionally stretch the sides of the dough out a bit to make it even thinner. Gently slide the pizza from the peel onto the heated baking stone. Bake for 7 to 8 minutes, or until the crust is golden and the cheese is bubbling and caramelized and the edges of the pizza are golden brown. Remove the pizza carefully from the oven with the pizza peel, transfer to a wooden cutting board or foil, drizzle the top with olive oil, some grated parmigiano-reggiano cheese, and chiffonade of fresh basil. Slice and serve immediately and/or prepare the second pizza.
If you’re serving two pizzas at once, I recommend placing the cooked pizza on a separate baking sheet while you prepare the other pizza. In the last few minutes of cooking, place the prepared pizza into the oven (on a rack below the pizza stone) so that it is extra hot for serving. Otherwise, I recommend serving one pizza fresh out of the oven, keeping the oven hot, and preparing the second pizza after people have gone through the first one! The pizza will taste great either way, but it is at its prime within minutes out of the oven!