Prepare a grill for medium heat (or preheat oven to 400°F). Combine vinegar, sugar, 1 teaspoon salt, and 2 tablespoons water in a small bowl. Add carrot and let sit until slightly softened, 15–20 minutes.
Meanwhile, mash anchovy and 1/2 garlic clove to a paste with the side of a chef’s knife. Transfer to a medium bowl and stir in carrot tops, basil, parsley, lemon juice, and 2 tablespoons oil.