Once you have a good sear on one side, turn the chicken pieces over and move them to the cooler side of the grill.
If you are using a gas grill, maintain the flame on only one side of the grill, and move the chicken pieces to the cooler side, not directly over the flame. Reduce the temp to low or medium low (between 250°F and 275°F, no more than 300°F).
Cover the grill and cook undisturbed for 20-30 minutes.
Turn the chicken pieces over and baste them with with your favorite barbecue sauce. Cover the grill again and allow to cook for another 15-20 minutes.
Repeat, turning the chicken pieces over, basting them with sauce, covering, and cooking for another 10-30 minutes.
The timing will depend on your grill set-up, the size of your chicken pieces, and how cold your chicken pieces are to start with.
The chicken is done when the internal temperature of the chicken pieces are 160°F for breasts and 170°F for thighs, when tested with a meat thermometer.
If you want, you can finish with a sear on the hot side of the grill. To do this, put the pieces, skin side down, on the hot side of the grill. Allow them to sear and blacken slightly for a minute or two.
Top with additional fresh BBQ sauce if desired and serve.