Soak the wood chips in water for at least 30 minutes.
In a medium saucepan over medium heat, cook the oil, onion, and garlic until golden, about 10 minutes, stirring often. Add the rest of the sauce ingredients and stir to combine. Bring the sauce to a simmer, reduce the heat to low, and cook until slightly thickened, 10 to 15 minutes, stirring often.
In a small bowl mix the rub ingredients. Sprinkle the rub evenly all over the chicken pieces.
Prepare the grill for direct and indirect cooking over medium heat (350° to 450°F).
Brush the cooking grates clean. Grill the chicken, skin side down, over direct medium heat, with the lid closed as much as possible, until golden brown, 8 to 10 minutes, turning occasionally. Move the chicken pieces over indirect medium heat. Drain and add the wood chips to the charcoal or to the smoker box of a gas grill, following manufacturer’s instructions. Continue to grill the chicken, with the lid closed, for about 20 minutes. Then brush both sides with a thin layer of the sauce and cook until the juices run clear and the meat is no longer pink at the bone, about 15 minutes, occasionally turning and brushing with the sauce. Serve warm or at room temperature with the remaining sauce on the side.