Pat the chicken dry and trim away excess fat while leaving the skin intact. Mix the olive oil, paprika, garlic powder, onion powder, kosher salt, black pepper, and optional chili powder or cayenne into a seasoning paste.

Rub the seasoning mixture evenly over every chicken piece, including beneath the edges of the skin when possible. For extra flavor and crispier skin, refrigerate the seasoned chicken uncovered or covered for 2–8 hours.

Prepare a two-zone grill with one side at medium-high heat, approximately 400–450°F, and the other side cooler for indirect cooking. If desired, thin the barbecue sauce with apple cider vinegar or water.

Place the chicken skin-side down over the cooler, indirect side of the grill. Cover and cook for approximately 25–35 minutes, turning occasionally, until the pieces are nearly done.

Move the chicken briefly over direct heat to brown and crisp the skin. Watch closely because the rendered fat can cause flare-ups.

Brush the chicken generously with barbecue sauce during the final 5–10 minutes, turning and brushing once or twice. Continue cooking until the thickest part reaches at least 165°F; thighs are often especially tender at 175–185°F.

Transfer the grilled chicken to a plate and rest for 5 minutes. Serve with the remaining barbecue sauce on the side.