Put the tomatoes in a large bowl. Heat the olive oil and garlic in a small saucepan over medium heat until the garlic is translucent, but not at all browned, 1 to 2 minutes. Stir in the crushed red pepper flakes, then pour the garlic oil over the tomatoes. Add 1 teaspoon salt and toss to coat.
Cook the pasta according to package directions. Transfer the pasta directly from the boiling water to the bowl with the tomatoes using a pasta fork or tongs. Sprinkle in the Parmesan and some of the basil and toss to coat; let stand 5 minutes so that the sauce absorbs slightly. Top with the remaining basil and a little additional olive oil and serve with additional Parmesan.