Meanwhile, light a charcoal grill or preheat a gas grill. A few minutes before the turkey has finished simmering, add 2 cups of the hickory chips to the coals. When the chips start smoking, brush the turkey breast with oil. Set the turkey, breast side down, on the grill. Cover and smoke over a low fire or flame for 15 minutes. Baste the turkey with the reserved braising liquid; turn it breast side up and baste again. Cover the grill and continue smoking the turkey for about 40 minutes longer, basting occasionally with the braising liquid and adding more coals or hickory chips to the grill as necessary. The turkey is done when an instant-read thermometer inserted in the inner thigh registers 165°. Transfer the turkey to a cutting board and let rest 20 minutes before carving.