2. Place half of the sauce in a shallow bowl, add the chicken breasts and marinate for 1 hour or overnight in the refrigerator. Reserve the remaining sauce separately.
3. To make the slaw, combine the red and green peppers, red onion, cabbage and dill in a medium bowl.
4. In a small saucepan, combine the chicken broth, 1/4 cup of the vinegar, oil, sugar and salt. Bring the mixture to a boil, pour it over the vegetable mixture and season with black pepper. Set the slaw aside for at least 1/2 hour or overnight in the refrigerator.
5. Preheat the grill to medium-high.
6. Remove the chicken from the marinade and place it on the hot grill. (Discard the marinade.) Grill the chicken on both sides, brushing from time to time with the reserved barbecue sauce, until the chicken is cooked through, about 6 minutes per side.
7. Serve the chicken along with a large spoonful of slaw.