In a large pot, boil 4 quarts water. Add the pasta and 1 tablespoon salt, then cook, stirring occasionally, until just shy of al dente. Reserve 1 cup of the cooking water, then drain. Return the pasta to the pot; set aside off heat.
In a 12-inch skillet over medium, heat the oil until shimmering. Add the tomatoes, sweet peppers, cubanelle pepper, garlic, tomato paste, pepper flakes, ½ teaspoon salt and ¼ teaspoon black pepper. Stir, then cover and cook, stirring occasionally, until the tomatoes burst and the peppers have softened, 8 to 10 minutes.
Stir in ¼ cup reserved pasta water, scraping up the browned bits, then add to the pasta along with another ¼ cup reserved pasta water. Cook over medium, stirring, until the pasta is al dente, about 2 minutes. If needed, adjust the consistency with additional reserved pasta water. Off heat, taste and season with salt and black pepper. Serve drizzled with additional oil and sprinkled with the basil and cheese.
