To make the brine: Combine the hot water, salt, and brown sugar in a large plastic container and stir to dissolve. Add the ice and make sure the brine is well-chilled. Add the chicken pieces; cover and refrigerate for 4 to 6 hours.
To make the cider spritz: Once the chicken has finished brining, combine water, vinegar, oil, hot sauce, Worcestershire sauce and salt in a food-safe spray bottle. Set aside.
Prepare a charcoal fire using about 6 pounds of charcoal and burn until the coals are completely covered with a thin coating of light gray ash, 20 to 30 minutes. Spread the coals evenly over the grill bottom, position the grill rack above the coals, and heat until medium-hot (when you can hold your hand 5 inches above the grill surface for no longer than 3 or 4 seconds). Or, for a gas grill, turn on all burners to high, close the lid, and heat until very hot, 10 to 15 minutes.
Apply some oil to the grill grate. Place the chicken on the grill, leaving plenty of space between each piece. Grill until seared, 1 to 2 minutes per side for legs and thighs, and 3 or so minutes for breasts. Move the chicken to medium-low heat or reduce the heat to medium; continue to grill, turning occasionally and squirting with the marinade, until the juices run clear when pierced and an instant read thermometer registers 165 degrees, about 35 minutes total. Remove the pieces from the grill as they cook and transfer to a warm platter. Give them a final squirt of sauce for flavor and serve immediately.