Whisk all the ingredients together in a medium bowl. Cover and refrigerate for at least 2 hours before using. Reserve a ramekin full of sauce, about 1/2 cup, to serve on the side.
For the chicken:
Preheat half of the grill to medium-high heat and the other half to low heat. Using tongs, dip a folded towel into olive oil and wipe over the grill grate.
Season the chicken pieces with salt and pepper. Arrange the chicken, skin side down, on the hottest area of the grill. Grill until seared and golden brown and crisp on each side, about 5 minutes. Move the chicken to the cooler part of the grill and brush with barbecue sauce. Continue to cook until the juices run clear, about 20 to 25 minutes. Do not forget to flip the chicken halfway through its cooking time. Transfer the chicken to a serving platter and serve with sauce on the side.