1. Preheat grill.
2. Warm the oil in a medium saucepan over medium-low heat. Add the onion and green pepper; cook 5 minutes, stirring, until softened. Stir in barbecue sauce. Cook for 3 minutes to warm through and blend flavors, stirring often.
3. Season chicken with salt and pepper. Combine the chicken and 1 cup of the sauce in a large zip-top plastic bag; flip to coat. Marinate in the refrigerator for 10 minutes or up to 2 hours.
4. Remove the chicken from the marinade; discard remaining marinade and bag. Place the chicken on an oiled grill over medium-hot coals; cook 12 minutes, turning once. Serve chicken with the remaining barbecue sauce.