Cover chicken breast with plastic wrap and pound gently with meat mallet or rolling pin until ½-inch thick.
Rub each piece of chicken with olive oil; season on all sides with BBQ rub (or kosher salt and freshly-ground black pepper to taste).
FOR A CHARCOAL GRILL: Open the bottom vent completely. Light the large chimney starter filled with charcoal briquettes. When the top coals are partially covered with ash, pour evenly over grill. Set cooking grate in place, cover, and open lid vent completely. Heat grill until hot, about 5 minutes.
FOR A GAS GRILL: Turn all burners to high, cover, and heat grill until hot, about 15 minutes. Leave all burners on high.
Oil the cooking grate. Place the chicken on the grill and cook (covered if using gas) until the chicken has nice grill marks, about 3-5 minutes. Flip the chicken over and continue to cook until the meat reaches an internal temperature of 165° F, about 3-5 more minutes. Baste with barbecue sauce during the final 3 minutes.
Transfer chicken to a cutting board and let rest for 5 minutes. Serve with additional sauce on the side.