Bring a large pot of water to a boil. Season aggressively with salt. Add the rigatoni and cook 3 minutes less than the package directions. Reserve 1/2 cup pasta water and drain.
Meanwhile, preheat the oven to 400 degrees F and line a baking sheet with parchment paper.
Toss the eggplant and tomatoes with 3 tablespoons olive oil and sprinkle with salt and pepper. Spread on the baking sheet and roast until the eggplant is golden brown and tomatoes are tender, 25 to 30 minutes. (Leave the oven on.)
Toss the rigatoni with the eggplant, tomatoes, about 1/4 cup pasta water and the marinara sauce in a large bowl. If it looks dry, add the remaining pasta water. Stir in 1 cup mozzarella.
Spray a 9-by-13-inch baking dish with nonstick spray. Transfer the pasta to the baking dish and top with the remaining cup mozzarella and the Parmesan and drizzle with the remaining tablespoon olive oil. Bake until the cheese is melted and golden brown, 20 to 25 minutes. (Turn the broiler on at the end if you want the cheese to brown more.) Garnish with additional Parmesan and the chopped basil and serve.