In a large covered pot, bring 4 quarts salted water to a boil. Lightly grease an 8x11-inch baking dish or 6 individual (1-1/2 cup) gratin dishes with 1 tsp. of the olive oil.
Add the penne or ziti to the boiling water and cook until it’s just tender but still firm to the tooth, about 11 minutes. Drain the pasta well and return it to its cooking pot.
Meanwhile, heat the remaining 2 Tbs. oil in a large straightsided skillet over medium-high heat. When the oil is hot, add the sausage, let it sit for a minute, and then start stirring and breaking it into bite-size pieces with the side of a slotted metal spoon. Cook until lightly browned, another 2 to 3 minutes. Transfer to a plate using the slotted spoon.
Add the garlic to the pan, season with salt, and cook, stirring constantly, until it colors slightly, about 30 seconds. Add the tomatoes and their juices and cook at a rapid simmer, stirring occasionally and breaking up the tomatoes with the spoon, for 5 minutes so the sauce thickens slightly.
Meanwhile, position an oven rack about 6 inches from the broiler element and heat the broiler on high. Stir the sausage and its juices, the diced peppers, and 1/3 cup of the Parmigiano into the sauce. Cook, stirring, until the sausage is cooked through, 3 to 5 minutes. Taste for salt and pepper. Pour the sauce over the cooked pasta in the pot and stir well. Spread the pasta and sauce evenly in the baking dish or gratin dishes. Sprinkle with the mozzarella and the remaining 1/3 cup Parmigiano. Put the baking dish or gratin dishes on a baking sheet and broil until the cheese melts and browns in places, 2 to 4 minutes. (check often to be sure they don’t burn). Serve immediately with more Parmigiano, if you like.