Mix 2 cups self-rising flour, sugar, and Parmigiano-Reggiano cheese for dough in a bowl until well combined. Add olive oil and Greek yogurt and mix with a spoon until a shaggy dough forms. Clean off the spoon and mix the dough with your hands until it mostly comes together.
Transfer dough to a work surface and continue pushing it together until it smooths out. It will be dry; don’t be tempted yet to add more liquid. Knead dough until smooth and elastic, 3 to 4 minutes, only adding a teaspoon or two of water or flour if absolutely necessary. Cut dough into two equal pieces.
Line a rimmed baking sheet with parchment paper and sprinkle 1 tablespoon cornmeal over the surface.
Roll one piece dough on a lightly floured surface to an 1/8-inch-thick rectangle. Transfer to the prepared baking sheet.
Top dough with ½ of the pizza sauce, spreading it almost, but not quite all the way, to the edges. Sprinkle ½ of the mozzarella over the sauce, then ½ of the Parmigiano-Reggiano over top. Arrange ½ of the pepperoni slices over the cheese, then drizzle 1 teaspoon oil over top.
Bake in the center of the preheated oven until crust is beautifully browned and toppings are bubbly, about 15 minutes. Transfer to a wire rack and let sit for 5 minutes before slicing and serving.
As soon as you remove the pizza from the oven, repeat Steps 3 to 6 to top and bake second pizza.