In a large container, whisk water, salt, brown sugar, peppercorns, and garlic until solids are dissolved.
Submerge chicken pieces in brine, cover, and refrigerate for 2 hours.
Remove chicken from brine, pat completely dry with paper towels, and let sit at room temperature for 15 minutes.
Preheat grill to medium-high heat (400°F) and oil the grates.
Place chicken skin-side down on the grill for 5 to 7 minutes until skin is crispy and browned.
Flip chicken and move to a lower heat zone; cook for another 10 to 15 minutes.
Whisk BBQ sauce, apple cider vinegar, and smoked paprika in a bowl.
Brush chicken with glaze during the final 5 minutes of cooking, flipping once to caramelize without burning.
Ensure internal temperature reaches 165°F (74°C) using a probe thermometer.
Rest chicken for 5 minutes before serving to allow juices to redistribute.