1. In a big pot of water brought to a boil, add salt and pasta, then cook until it's almost done (al dente), which is usually 2 to 3 minutes less than what's written on the package. Set aside 1 cup of the pasta water, then drain the rest.
2. While the pasta cooks, in a large skillet with straight sides over medium heat, heat the oil. Add the garlic and cook for about 3 minutes, stirring often, until it becomes soft and a light shade of gold. Add the tomato paste and cook for about a minute, stirring, until it turns a brick-red color. Add the tomatoes, red pepper flakes, and a half teaspoon of salt. Cook for about 5 minutes, stirring occasionally, until tomatoes are soft and starting to break apart.
3. Turn the heat up to high and put in vinegar and sugar. Let it boil for a minute before adding pasta, Parmesan, and three-quarters of the cup of reserved pasta water. Keep cooking and stirring for about 2 more minutes, until the sauce becomes thick and sticks to the pasta. If needed, add more reserved pasta water by the tablespoon to thin out the sauce so it coats the pasta.
4. Put the pasta in bowls and divide it equally. Top with Parmesan, red pepper flakes, and basil.