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 both cans of the tomatoes and 1 cup 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 basil and stir in the chopped basil. Season with salt and pepper.
While the sauce simmers, heat the remaining 1 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 salted boiling water until very al dente, about 2 minutes less than the package directions instruct. Drain the pasta and add it to the large bowl.
Add the tomato sauce and half each of the mozzarella, provolone and Parmesan cheeses to the bowl and stir well to combine. Transfer the pasta-sausage mixture to a 3- to 4-quart baking dish and top with the remaining mozzarella, provolone and Parmesan cheeses. Bake, uncovered, until browned, about 15 minutes. Let the pasta rest about 10 minutes before serving.
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