For the brine and chicken: Combine the salt, brown sugar and 16 cups (1 gallon) cold water in a large plastic container and stir to dissolve. Add the chicken; cover and refrigerate to marinate for 4 to 6 hours.
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.
For the basting liquid: Whisk together 1/2 cup water, the vinegar, oil, Worcestershire sauce and hot sauce in a medium bowl. Set aside.
Drain the chicken well and pat dry; sprinkle with pepper. 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. Reduce the heat to medium or move the chicken to a cooler part of the grill. Continue to grill, turning occasionally and brushing with the basting liquid, until the juices run clear when the meat is pierced, 18 to 20 minutes. Remove the pieces from the grill as they cook and transfer to a warm platter. Serve immediately!