Add vinegar, ketchup, brown sugar, Worcestershire, mustard, salt, pepper and garlic powder to a medium sauce pan.
Stir to incorporate over medium-high heat and once sauce starts to bubble, reduce heat and simmer 20 minutes.
As sauce is simmering, heat grill to medium or start grill coals. If using coals, place coals off center to grill with indirect heat. If baking or broiling, prepare broiler pan with cooking spray.
After sauce has reduced for 20 minutes, sprinkle chicken breasts with salt, pepper and garlic powder on both sides.
Pour approximately 1/3 cup sauce into a small bowl and set aside to serve for dipping sauce.
For grilling: Heat chicken on grill until the meat is able to be turned, approximately 3 minutes. Turn breasts and with a basting brush, generously apply sauce to side that just cooked. After another 3 or 4 minutes, turn chicken and brush the other side. Repeat this process a couple of times until chicken reaches 160 degrees internal temperature, applying the sauce as generous as you like and I like it pretty generous. I turned the chicken about 4 times, in total. Total cooking time was about 20 minutes. (If this is too much time turning, you can definitely just turn and sauce a couple of times. I wanted to get a lot of sauce on my chicken.)
For broiling: Place oven rack to lower third in oven. Preheat on High Broil for a couple of minutes. Liberally brush sauce over top of chicken and broil for 8-10 minutes, watching that sauce doesn't start to burn. Add more sauce if it seems to become dry. Turn chicken and brush again. Continue to watch chicken, brushing more sauce for more flavor and broil until chicken reaches 160 degrees and juices run clear.
For baking: Preheat oven to 400 degrees, liberally brush sauce over top of chicken and bake for 10 minutes. Turn chicken and brush liberally again with sauce and bake another 10-15 minutes until internal temperature is 160 degrees.
Serve with extra sauce for dipping.