Make the sauce and rub
In a medium saucepan, boil the apple juice and vinegar over medium-high heat until reduced by half, 5 to 7 minutes. Lower the heat to medium low and add the remaining sauce ingredients. Simmer for 5 minutes, stirring occasionally. Season to taste with salt and pepper and set aside.
In a small bowl, mix the rub ingredients.
Grill the chicken
Trim any excess fat from the chicken, and season the chicken evenly with all of the rub.
Prepare the grill for indirect cooking over medium-low heat (300°F to 350°F): For a charcoal grill, ignite a chimney starter full of lump charcoal and burn until the edges of the coals look ashy, about 15 minutes. Carefully dump out the coals and use tongs to spread them over half of the charcoal grate. Put the cooking grate in place, close the lid, and let the coals burn down to about 350°F. For a gas grill, turn all the burners on high and close the lid to heat the grill. Then turn off all but one burner to create a cool zone and adjust that burner to maintain a temperature of 300°F to 350°F.
Brush the cooking grate clean. Scatter the wood chips over the lit charcoal, or for a gas grill put them in a smoker box, following the manufacturer’s instructions.
When you see smoke pouring out of the grill, arrange the chicken pieces skin side up over the cooler part of the grill. Close the lid and smoke the chicken for 45 minutes. (For a charcoal grill, if the temperature drops below 300°F, remove the chicken and scatter a layer of new coals on top of the old ones.)
Move the chicken to the hot side of the grill, and grill, uncovered, turning and moving the chicken regularly to manage flare-ups, until lightly charred on both sides, 4 to 5 minutes.
Move the chicken back to the cooler side of the grill, brush with a thin layer of sauce, close the lid, and cook until an instant-read thermometer inserted in a thick part of the chicken registers 170°F, 10 to 15 minutes, brushing occasionally with more sauce. (The drumsticks usually take a little longer than the thighs.) Serve hot or at room temperature, with the remaining sauce on the side.
You can make the sauce up to a week ahead. Refrigerate in a covered container.