Make sauce: Combine ingredients in a medium saucepan. Cover and simmer 45 minutes to blend flavors, stirring occasionally with a long spoon (sauce spatters). Add a little water if sauce gets too thick to pour. Let cool, then transfer to a plastic container and chill up to 1 week.
Make slaw: Toss vegetables in a large bowl to blend. Transfer to a resealable plastic bag. Put remaining ingredients in a small container with a tight-fitting lid. Chill slaw and dressing up to 2 days.
Prepare chicken: In a large bowl, coat chicken in oil; pack in a resealable plastic bag and chill up to 2 days, or freeze.
Build a charcoal or wood fire in a grill and let burn to medium (350° to 450°; you can hold your hand 5 in. above cooking grate only 5 to 7 seconds). Bring salad dressing to room temperature. Set aside 1/2 cup sauce.
Grill chicken until browned all over, about 15 minutes, turning occasionally. Turn again, generously brush tops with some of remaining sauce, and cook a few minutes; repeat turning and brushing 2 more times, until chicken is well-browned and cooked through, 10 to 15 minutes total.
Toss slaw and dressing in a large bowl and serve with chicken and reserved sauce.
Note: Nutritional analysis is per serving.