In a small bowl, mix all the dry ingredients. Set aside.
Build a low direct fire. Spread an even layer of unlit charcoal in the bottom of the grill. Fill a chimney one-third full with charcoal. Stuff two sheets of newspaper in the bottom of the chimney and light it. When the coals are fully engaged – you should see flames peeking over the top – pour them over the unlit charcoal. If using a gas grill, light the gas and adjust temperature on both sides to low.
While the fire is coming to temperature, place the chicken thighs on a baking sheet, skin side up. Sprinkle the chicken with half of the dry rub.
When you can hold your hands over the fire for no more than 8-10 seconds, clean the grill grate. Set the chicken thighs, skin side down, on the grates. Sprinkle the remainder of the dry rub on the chicken. Cook, covered, for 30-40 minutes, checking periodically to avoid excessive charring on the skin. Rotate thighs over the fire if there are hot spots. If using a charcoal grill, adjust the vents to maintain a temperature of 300oF/149oC.
Flip chicken and continue to cook until a thermometer inserted into the thickest part of the meat registers 170oF/77oC, about 15-20 minutes more. Transfer chicken to a platter and let it rest for 10 minutes. Slather with Alabama White Sauce and serve.