In the bowl of a stand mixer fitted with a dough hook, combine flour, 2 teaspoons salt, and yeast. Mix until combined. Add water and olive oil. Dough will be tacky and should stick to the sides of the bowl. Continue mixing until well incorporated and smooth, adding 1-2 tablespoons of water if necessary, to make dough cohesive and smooth.
Scrape dough into a large, oiled bowl and cover with plastic wrap. Let stand at room temperature for 8 hours and up to 24.
Grease two 10-inch cast iron skillets and/or round cake pans with 1 tablespoon olive oil. Flour a work surface and scrape dough out of bowl. Divide dough into 2 pieces. Using lightly floured hands, form each into a ball, place into the skillets, and turn to coat in oil. (Alternatively, keep one dough wrapped while the other rises; repeat with remaining dough after first pizza bakes.) Using fingers, press the dough to the edges of the pan. (Don’t worry about stretching the dough to fill the entire pan; dough will do so naturally during the next rise.) Cover with plastic wrap and let rise in a warm place for 1 hour, until dough looks puffed and begins to fill the skillet.
Preheat oven to 550 degrees F with racks in the upper and lower thirds. Remove plastic wrap. Lightly oil the top of dough and use your fingertips to gently press dough to fill the pan evenly. Do not disturb any air pockets that have formed in the dough. Divide pizza sauce, mozzarella, basil, and red pepper flakes between both dough rounds. Drizzle with olive oil and season with salt and pepper.
Transfer to oven and bake until pizza crusts are deeply golden and cheese is bubbling about 15 minutes. Sprinkle with Parmigiano-reggiano and top with additional basil leaves. Cut into slices and serve.