Preheat the oven to 450 degrees F. Bring a large pot of salted water to a boil.
Heat 2 tablespoons of the oil in a medium saucepan over medium heat. Add the garlic and red pepper flakes and cook, stirring, until the garlic is golden, about 2 minutes. Add the tomatoes and 1 cup of water. Increase the heat to medium high and bring to a simmer. Reduce the heat slightly, add the basil sprigs and simmer rapidly, uncovered, until thickened, 15 to 20 minutes. Discard the basil and season with salt and pepper.
While the sauce simmers, heat the remaining tablespoon oil in a large skillet over medium-high heat. Add the sausage and cook, breaking up any clumps with a wooden spoon, until brown, about 5 minutes. Transfer the sausage to a large bowl with a slotted spoon.
Cook the rigatoni in the boiling water until very al dente, about 2 minutes less than the package directions. Drain the pasta and add it to the bowl.
Add the tomato sauce, ham, pepperoni, bacon, half of the mozzarella and half of the Parmesan to the bowl and stir well to combine. Transfer the pasta to a 3- to 4-quart baking dish and top with the remaining mozzarella and Parmesan. Bake, uncovered, until browned, about 15 minutes. Let the pasta stand about 10 minutes before serving.
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