Season chicken with salt and 1/2 tsp. pepper. Sprinkle with granulated garlic. Cover and refrigerate until ready to grill.
Prepare a grill for a two-zone fire. (For a gas grill, preheat one burner to medium-high heat and the rest of the burners to medium-low; for a charcoal grill, bank two thirds of the coals to one side of the grill and spread the remaining coals over the other side.) Clean and oil the grill grates.
Stir together brown sugar, vinegar, mustard, ketchup, paprika, and the remaining 1/4 tsp. pepper in a small bowl.
Place the chicken (skin sides down for the thighs) on the cooler side of the grill. Cover and cook, turning occasionally and switching positions (so all pieces cook evenly), until the chicken is cooked through (the internal temperature should read 165°F on an instant-read thermometer), 15 to 30 minutes. Uncover and transfer the chicken to the hotter part of the grill. Continue cooking, basting chicken with the glaze and turning often, until lightly charred and caramelized, 4 to 5 minutes. Transfer to a platter and let rest 5 to 10 minutes before serving.