Preheat the oven to 450 degrees F. Bring a large pot of salted water to a boil.
Heat 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 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 and season with salt and pepper.
Cook the rigatoni in the salted boiling water until very al dente, about 2 minutes less than the package directions. Drain the pasta and transfer it to a large bowl.
Add the tomato sauce, chicken, Asiago and Parmesan to the bowl and stir well to combine. Transfer the pasta to a 3- to 4-quart baking dish and top with the mozzarella and the pecorino. Bake, uncovered, until browned, about 15 minutes. Let the pasta stand about 10 minutes before serving.
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