You are sure to love this irresistible pie that combines 2 favorites: vodka pasta and homemade pizza.
Place a pizza stone or an overturned rimmed baking sheet in oven, and preheat oven to 500°F. (Do not remove pizza stone while oven preheats.)
Cook pancetta in a large, high-sided skillet over medium, stirring occasionally, until browned and crisp, about 7 minutes. Add 3 tablespoons of the olive oil to pancetta, and reduce heat to medium-low. Smash 5 of the garlic cloves, and add to pancetta. Stir in crushed red pepper, and cook, stirring often, until garlic is lightly browned and fragrant, about 5 minutes. Carefully add tomatoes, and stir to combine. Cook, stirring occasionally and crushing tomatoes with a potato masher, until sauce thickens slightly, about 8 minutes. Add vodka, and cook 2 minutes. Add peas, and stir to combine; cook until heated through, about 1 minute. Stir in cream and 1 teaspoon of the salt. Set aside.
Cook pasta 1 minute less than package directions for al dente. Drain.
Finely chop remaining 3 garlic cloves on a cutting board, and sprinkle with remaining pinch of salt. Mash garlic and salt into a paste; transfer to a small saucepan. Add butter and remaining 2 tablespoons olive oil. Heat gently over medium-low until butter is melted and garlic begins to sizzle, about 5 minutes. Remove from heat.
Stretch dough into a 14-inch circle on parchment paper. Brush dough with garlic butter until lightly covered, concentrating on edges of dough round; reserve remaining garlic butter. Slide pizza crust, with parchment paper, onto hot pizza stone.
Bake in preheated oven until lightly golden, about 5 minutes. Remove from oven; brush with reserved garlic butter, and sprinkle evenly with mozzarella, leaving 1/2-inch border. Toss cooked pasta with vodka sauce, and spread over mozzarella. Sprinkle with 1/2 cup of the Parmigiano-Reggiano, and return to oven.
Bake at 500°F until crust is browned and cheese is melted, 5 to 10 minutes. Sprinkle with parsley and remaining Parmigiano-Reggiano. Cut into 8 slices.