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 sprigs and simmer rapidly, uncovered, until thickened, 15 to 20 minutes. Discard the basil sprigs and stir in the chopped basil and parsley. 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 artichokes to the skillet and cook, stirring occasionally, until hot, about 3 minutes. Add the artichokes 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, half of the fontina 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 fontina and Parmesan. Bake, uncovered, until browned, about 15 minutes. Let the pasta stand about 10 minutes before serving. Sprinkle with chopped parsley.
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