Step 1

In a large pot, bring 2 quarts water to a boil. Add 1/2 tablespoon salt and the spaghetti, then cook, stirring occasionally, for 5 minutes. Reserve 2 cups of cooking water, then drain and set aside.

Step 2

In a 12-inch skillet over medium, heat 1 tablespoon of oil and garlic cloves, then cook, stirring often, until the garlic is light golden brown, about 1 minute. Off heat, remove and discard the garlic. Add the pepper flakes, both types of olives and the capers, then cook over medium-high, stirring, until the capers begin to brown, about 1 minute. Add the tomatoes and cook, stirring occasionally, until most of the liquid has evaporated, 5 to 7 minutes.

Step 3

Add the reserved tomato juice and 1 cup of the reserved cooking water; bring to a simmer. Add the pasta and toss to coat. Cover and cook, tossing occasionally, until the pasta is al dente and the sauce clings lightly to the noodles; add more cooking water if needed.

Step 4

Remove from the heat, cover and let stand for 3 minutes. Stir in the basil, cheese and remaining 1 tablespoon olive oil. Taste and season with salt and black pepper. Serve topped with additional cheese.