Mix the sugar, salt and 3 quarts (12 cups) water together in a large plastic or stainless steel vessel. Add the chicken quarters to the mixture and let brine, refrigerated, for at least 2 hours.
Set a wire rack on a baking sheet. Remove the chicken from the brine, pat dry with paper towels and then set on the wire rack for at least 1 hour at room temperature, though it is ideal to do this step overnight in the refrigerator. (If you decide to refrigerate it overnight, be sure to bring the chicken to room temperature for about an hour before grilling.)
Heat a grill to medium heat (about 350 degrees F). In a large bowl, toss the chicken with the cornstarch and vegetable oil.
Grill the chicken skin-side down for 15 minutes, keeping an eye on it to make sure the skin doesn't burn. Flip the chicken and cook until it reaches an internal temperature of 165 degrees F, about 20 minutes. Remove and let rest for at least 10 minutes. Serve with Alabama White BBQ Sauce.
Alabama White BBQ Sauce:
Stir together the mayonnaise, vinegar, horseradish, mustard and pepper in a medium bowl. Season with salt. Serve with grilled chicken.