Preheat oven to 450 degrees F. Bring a large pot of salted water to a boil.
Heat 2 tablespoons of oil in a medium saucepan over medium heat. Add the garlic and cook, stirring, until golden, about 2 minutes. Add both cans of the tomatoes and 1 cup of water. Increase the heat to medium-high and bring to a simmer. Reduce the heat slightly and simmer rapidly, uncovered, until thickened, 15 to 20 minutes. Stir in the ricotta and season with salt and pepper.
While the sauce simmers, heat the remaining tablespoon oil in a medium skillet over medium heat. Add the pepperoni and cook, stirring occasionally, until it just starts to crisp, about 4 minutes. Add the spinach and stir, breaking up the clumps, just to warm. Transfer to a large bowl.
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, half each of the Asiago and pecorino to the bowl and stir well to combine. Transfer the pasta to a 3- to 4-quart baking dish and top with the remaining Asiago and pecorino. Bake, uncovered, until browned, about 15 minutes. Let the pasta rest for about 10 minutes before serving.
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