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 tomatoes and 1 cup of water. Increase the heat to medium-high and bring to a simmer. Reduce the heat slightly, add the basil and simmer rapidly, uncovered, until thickened, 15 to 20 minutes. Discard the basil. Season with salt and pepper.
Meanwhile, 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. Add the mushrooms to the skillet and cook, stirring occasionally and reducing the heat slightly if necessary, until golden brown, about 6 minutes. Add the mushrooms to the bowl with the sausage.
Cook the rigatoni in the salted 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 and half of the provolone and pecorino to the bowl and stir well to combine. Transfer the pasta to a 3- to 4-quart baking dish and dollop spoonfuls of the ricotta on top. Top with the remaining provolone and pecorino. Bake, uncovered, until browned, about 15 minutes. Let the pasta stand about 10 minutes before serving. Sprinkle with chopped basil.
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