Sungold Tomato Pasta

1 1/2 teaspoons kosher salt, plus more for pasta water1 pound pasta, any shape1/4 cup olive oil, plus more to finish2 cloves garlic, smashed and peeled27 ounces (about 4 cups) sungold or cherry tomatoes, halved1 cup freshly grated Parmesan, plus more to finish1 cup freshly grated Pecorino Toscano cheese, or other mild sheep'smilk cheese1 cup chopped basilBring a large pot of water to a boil over medium-high heat. Season well withsalt. 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 garlicand cook until the garlic is fragrant and just beginning to brown. Add thetomatoes and the salt and cook, stirring often, until the tomatoes are justbeginning to break down and release their juices, about 3 minutes. Add the pastato the tomatoes and sprinkle with the Parmesan. Toss well to coat the pasta in thetomatoes. 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 ofolive oil and more Parmesan if desired.