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 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.
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, artichokes, red peppers, sopressata, olives, half of the mozzarella 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 mozzarella and Parmesan. Bake, uncovered, until browned, about 15 minutes. Let the pasta stand about 10 minutes before serving. Sprinkle with parsley.
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