Dissolve salt in 2 quarts cold water in large container. Submerge drumsticks in brine, cover, and refrigerate for 30 minutes to 1 hour.
Place spice rub on plate. Remove drumsticks from brine and pat dry with paper towels. Holding 1 drumstick by bone end, press lightly into rub on all sides. Pat gently to remove excess rub. Repeat with remaining drumsticks.
For a charcoal grill: Open bottom vent halfway. Light large chimney starter filled with charcoal briquettes ( 6 quarts). When top coals are partially covered with ash, pour evenly over half of grill. Set cooking grate in place, cover, and open lid vent halfway. Heat grill until hot, about 5 minutes.
For a gas grill: Turn all burners to high, cover, and heat grill until hot, about 15 minutes. Leave primary burner on high and turn off other burner(s). (Adjust primary burner [or, if using 3-burner grill, primary burner and second burner] as needed to maintain grill temperature between 325 and 350 degrees.)
Clean and oil cooking grate. Place drumsticks, skin side down, on cooler side of grill. Cover and cook for 25 minutes. Rearrange pieces so that drumsticks that were closest to edge are now closer to heat source and vice versa. Cover and cook until drumsticks register 185 to 190 degrees, 20 to 30 minutes.
Move all drumsticks to hotter side of grill and cook, turning occasionally, until skin is nicely charred, about 5 minutes. Transfer to platter, tent with aluminum foil, and let rest for 10 minutes before serving.