1. In a medium bowl, whisk together 1 cup room-temperature water and the yeast. Let stand 15 minutes at room temperature.
2. In bowl of a stand mixer using dough hook, mix 2 cups flour, 2 tsp. salt, 1 tbsp. olive oil, and yeast mixture on medium speed until well incorporated, scraping down sides of bowl as needed. Add remaining 1/2 cup flour and mix about 30 minutes, or until dough is very smooth and elastic.
3. Meanwhile, make tomato sauce: In a medium pot, heat 2 tbsp. olive oil over medium heat until hot but not smoking. Add garlic and swirl in hot oil until it starts to smell good, about 15 seconds. Stir in tomatoes and simmer, uncovered, to thicken and cook off the “canned” flavors, at least 25 minutes.
4. While sauce is cooking, put 1/4 cup olive oil and basil leaves in a food processor and whirl to finely chop basil, scraping down sides of bowl as needed. Stir 1 tbsp. basil oil into tomato sauce as it’s cooking, along with 1 pinch salt. Pour remaining basil oil into a small bowl and cover surface with a thin layer of olive oil.
5. As soon as dough is ready, divide into 4 portions. Using both hands, roll each portion with a circular pressing motion until it becomes a tight ball. Dust each ball with flour, set it on a floured baking sheet, and cover loosely with plastic wrap. Let dough rise 1 hour, or until almost doubled in size (do not let rise longer than 1 hour).
6. Put a pizza stone or baking sheet on bottom shelf of oven and preheat to 550° (or as high as oven will go), at least 25 minutes. Working with 1 dough ball at a time, set on a well-floured pizza peel or baking sheet and stretch into a 10- to 11-in. circle. To stretch into a 10- to 11-in. circle, first tap down center of ball with your fingertips to gently deflate it. Next, push it outward from the center with your fingertips. Then pick up the dough circle and, holding it under the rim, turn it like a steering wheel, letting the gravity of the dough help it stretch. Drape the dough over the backs of your hands and gently stretch outward, rotating periodically. Flop the stretched dough down onto the pizza peel.
7. Spoon 3 to 4 tbsp. tomato sauce onto dough, leaving at least a 1/2-in. border.
8. Plant tip of pizza peel (or long edge of baking sheet) on pizza stone (or preheated sheet) and shove pizza quickly onto stone and bake 3 to 6 minutes, or until crust looks dryish but not browned. Remove pizza from oven and sprinkle on about 1/2 cup mozzarella cubes in clusters, then 1 generous pinch oregano. Return to oven and cook 2 to 5 minutes more, or until crust is golden brown and firm but not rock hard. Transfer pizza to a cutting board and drizzle with basil oil. Assemble and bake rest of pizzas the same way.
Make ahead: Dough can be formed into balls (step 5) and chilled overnight, tightly covered with plastic wrap, instead of rising on counter (it will rise slowly in the fridge). You can also freeze the dough for up to 2 weeks (let chilled or frozen dough come to room temperature before stretching).