Place chicken skin side down, and pound it with a mallet to flatten slightly.
Combine stuffing mix, onion, butter, and 1/4 tsp ginger. Spoon stuffing mixture in a strip along center of each breast. Place apricot halved on top of stuffing. Wrap chicken around filling; tie each chicken roll with a string every 2 inches. Barbecue on rack about 5 1/2 inches above medium-hot coals 15 minutes, turning once or twice. Mix apricot jam, vinegar, and remaining 1/4 tsp ginger.
Brush jam mixture over chicken rolls; continue cooking until done, about 5 to 10 minutes.
