In a medium pan over medium heat, add 3 tablespoons olive oil, chopped garlic, rosemary sprig, and salt. Sauté gently until garlic has softened and rosemary has released its fragrance, about 5–6 minutes. Do not let it get to a boil! Add vinegar and bring to a simmer. Reduce the heat to medium-low and let the sauce reduce to ⅓ of the original volume. Stir occasionally.
Remove the olive oil mixture from the heat and discard the rosemary sprig. Season with salt and pepper, to taste. Finish with 1 tablespoon extra virgin olive oil, stirring in to form a loose paste. Keep warm until ready to use.
