In a nonstick 12-inch skillet over medium, combine 2 tablespoons of oil, the anchovies and garlic. Cook, stirring occasionally and breaking up the anchovies, until the garlic is light golden brown, 2 to 3 minutes. Stir in the pepper flakes and tomatoes, then cover and cook, stirring occasionally, until most of the whole tomatoes have burst, 5 to 7 minutes.

Use a fork or potato masher to gently mash the tomatoes. Off heat, stir in the vinegar and remaining 2 tablespoons olive oil. Taste and season with salt and pepper. Stir in the basil.