How to Make It
Whisk together buttermilk, water, salt, pepper, 1/2 cup of the hot sauce, and 1/4 cup of the brown sugar in a large bowl until sugar is dissolved; stir in lime slices and garlic.
Place buttermilk mixture and chicken in a large ziplock plastic freezer bag; seal bag, and chill 24 hours. Remove chicken from marinade, discarding marinade; pat dry with paper towels.
Heat a grill to medium (350°F to 450°F) with an area cleared of coals (for charcoal grill) or a burner turned off (for gas) to make an indirect heat area. Place chicken, skin side up, over indirect heat, and grill, covered, until a meat thermometer inserted in thickest portion registers 165°F, about 1 hour and 15 minutes.
Whisk together ketchup, butter, vinegar, and remaining 3 tablespoons hot sauce and 2 teaspoons brown sugar.
Brush ketchup mixture onto both sides of chicken; transfer, skin side down, to direct side of grill, and grill, uncovered, 1 to 2 minutes or until skin is browned and crispy; turn and grill, uncovered, 1 to 2 more minutes. Let rest 5 minutes before serving.
Allow 24 hours for brining.