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For the ravioli: Combine the cream, pecorino, ricotta and goat cheese in a large bowl and mix well. Season with the smoked sea salt and set aside. Lay out half of the pasta sheets and sprinkle with the smoked paprika. Drop 3/4 ounce piles of the cheese filling onto the sheets about 2 inches apart. Brush the egg wash around the cheese filling piles and top with the remaining pasta sheets. Seal the edges of the ravioli by pressing down along each seam. Using a decorative cutter or knife, cut the ravioli into 2 1/2-inch squares and place on a baking sheet. Refrigerate until firm, at least 1 hour or up to overnight.
When ready to use, bring a large pot of salted water to a boil. Add the raviolis and cook until the pasta is tender, 5 to 6 minutes. Reserve for topping on the pizza.
For the pizza: Place a pizza stone in the oven and preheat the oven to 550 degrees F. Toss the dough in the breadcrumbs and stretch to 14 inches in diameter. Layer the sliced mozzarella on the dough right to the edges. Toss 13 to 15 of the ravioli in the vodka sauce, and then cover the pizza with vodka sauce and place the raviolis on top. Place the smoked mozzarella over the top and bake on a pizza stone until cooked through and the cheese melts, 6 to 8 minutes. Garnish with the basil and grated pecorino to serve.