Bring a large pot of water to a rolling boil. Add 1 tablespoon salt, then the angel hair pasta. Cook according to package directions until al dente (usually 2–3 minutes). Just before draining, reserve 1 cup of pasta cooking water. Drain and set aside.

While the pasta cooks, heat the olive oil in a large skillet over medium heat. Add the sliced garlic and red pepper flakes (if using). Cook, stirring frequently, until the garlic is fragrant and pale golden, about 1–2 minutes. Add the halved cherry tomatoes, salt, and pepper. Cook, stirring occasionally, until the tomatoes soften and begin to burst, creating a light sauce, about 6–8 minutes.

Add the drained pasta to the skillet with the tomatoes. Toss well with tongs, adding a splash of reserved pasta water as needed to create a silky coating. Turn off the heat.

Gently fold in the torn basil and grated Parmesan. Divide among warm bowls, top with extra Parmesan and a drizzle of olive oil if desired, and serve immediately.