1. Combine 1 tablespoon salt and 1/2 teaspoon pepper. Sprinkle mixture evenly on chicken, then set chicken on a rimmed baking sheet. Let stand uncovered 1 hour.

2. Meanwhile, bring ketchup, vinegar, sugar, Worcestershire sauce, and garlic to a simmer in a small saucepan. Cook, stirring occasionally, until slightly thickened and reduced to 1 cup, about 4 minutes. Season with salt and pepper, and transfer sauce to a bowl.

3. 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.

4. Uncover grill; transfer chicken to direct heat. Brush with sauce and cook, turning frequently, until caramelized and lightly charred, about 3 to 4 minutes. Brush on more sauce as pieces come off the grill, or serve it on the side.