Make Sheet-Pan Pizza Dough. During the final rise, put a pizza stone or inverted baking sheet in the lower third of the oven and preheat to 500˚ for 1 hour.
Meanwhile, make the vodka sauce: Heat the olive oil in a medium saucepan over medium-high heat. Add the onion and cook, stirring, until softened, 5 minutes. Add the garlic and red pepper flakes and cook, stirring, until softened, 30 seconds. Add the tomato paste and cook, stirring, 1 minute. Add the tomatoes and 1/2 teaspoon salt. Bring to a boil, then reduce the heat to a simmer and cook, stirring, until thickened, 12 to 18 minutes. Stir in the heavy cream and simmer until slightly thickened, 3 to 5 minutes. Carefully add the vodka; cook 1 minute. Season with salt.
Puree the sauce in a blender. Remove 1 cup sauce to a bowl; set aside for serving. Refrigerate the remaining sauce until cooled.
Sprinkle the pecorino and parmesan over the dough. Top with the mozzarella, leaving a 1/2-inch border. Spoon the cooled vodka sauce over the top. Sprinkle with more grated cheese.
Put the baking sheet directly on the hot stone and bake until the crust is browned around the edges and the cheese is bubbling, 15 to 20 minutes. Let cool slightly.
Loosen the edges of the pizza with an offset spatula, then loosen the bottom. Slide the pizza onto a cutting board; cut into squares. Top with basil and more grated cheese. Serve with the reserved sauce.