For sauce, in a small saucepan combine brown sugar, onion, vinegar, mustard, celery seed, and garlic powder. Bring to boiling, stirring until the sugar dissolves. Remove from heat; set aside.
Skin chicken, if desired. In a small mixing bowl combine paprika, turmeric, and salt; rub over the chicken.
Grill chicken on an uncovered grill directly over medium coals for 20 minutes. Turn chicken; grill for 15 to 20 minutes more or until chicken is tender and no longer pink. (Or, place chicken on the unheated rack of a broiler pan. Broil 5 to 6 inches from the heat for 28 to 32 minutes, turning once.) Brush with sauce during the last 5 minutes of grilling or broiling.
Prepare sauce up to 48 hours ahead; cover and chill.