Bring a large pot of water to a boil over medium-high heat. Season well with salt. Add the pasta and cook for 3 minutes less than package directions, about 5 minutes. Drain well, reserving 1/2 cup of the pasta water.
Meanwhile, heat a large skillet over medium-high heat. Add the oil and garlic and cook until the garlic is fragrant and just beginning to brown. Add the tomatoes and the salt and cook, stirring often, until the tomatoes are just beginning to break down and release their juices, about 3 minutes. Add the pasta to the tomatoes and sprinkle with the Parmesan. Toss well to coat the pasta in the tomatoes. Sprinkle in the pecorino and basil and toss one more time to combine. Loosen the sauce with the reserved pasta water if needed. Serve with a drizzle of olive oil and more Parmesan if desired.