A salty-sweet vinegar sauce is brushed over this spice-rubbed chicken as it comes off the grill for next-level flavor.
Combine paprika, mustard, sugar, cayenne, 1 tablespoon salt, and 1/2 teaspoon black pepper. Sprinkle mixture evenly on chicken, then set chicken on a rimmed baking sheet. Let stand uncovered 1 hour.
Separately, combine vinegar, honey, and chile.
Heat a charcoal grill, creating a two-zone fire with vents partially open. Position coals on one side of the grill for direct high heat, and leave the other side empty for indirect lower heat. (On a gas grill, turn one side on high, and leave the other burners off.) Lightly coat chicken with olive oil and place over indirect heat. Cover and cook, turning occasionally, until pieces are cooked through and a thermometer inserted into thickest parts (not touching bones) registers 165 degrees, about 30 minutes.
Uncover grill; transfer chicken to direct heat. Brush on vinegar sauce as pieces come off the grill, and serve it on the side.