Preheat the oven to 325 F.
Rinse the turkey inside and out. Pat dry with paper towels. Place the turkey breast-side up on a rack in a roasting pan. Rub the turkey with the oil, a sprig of rosemary and garlic cloves. Place all of the rosemary and garlic inside the bird. Loosely tie the legs together. Place into the middle of the oven.
When the skin is light (after about 1 1/2 hours) cover the breast with a tent of foil to prevent overcooking the breast. Check the doneness after the bird has roasted about 3 to 3 1/2 hours. Turkey is done when the thigh is pierced deeply and juices run clear (180 to 185 F) or when the breast muscle reaches 170 to 175 F.
Remove the turkey from the oven. Let it stand about 20 minutes to allow juices to settle in the meat. Deglaze the pan by adding 1/2 cup water. Stir to scrape up browned bits. Pour pan drippings into a gravy separator. Reserve 1 cup of defatted pan drippings for sauce.
To make the sauce, in a pan or microwave dish, stir together the vinegar, defatted pan drippings and brown sugar. Warm the sauce until the flavor is just released, about 15 seconds in the microwave, or over low heat on the stove until steam "escapes." Don''t boil.
Carve the turkey and drizzle turkey slices with the warmed brown sugar sauce. Serve immediately.