Combine paprika, cayenne, garlic powder, onion powder, salt, and black pepper in a small bowl to create the dry rub.
Pat the chicken drumsticks dry with paper towels to remove excess moisture, which helps achieve crispy skin.
Rub the spice mixture evenly all over each drumstick, pressing it into the skin for full coverage.
Drizzle olive oil over the seasoned drumsticks, then massage it into the skin to enhance browning and crispness.
Preheat the grill to medium-high heat until hot and slightly smoky, about 10 minutes, to prepare for grilling.
Place the drumsticks onto the grill, spacing them evenly, and listen for a slight sizzle to confirm direct contact with the hot grates.
Grill the chicken for about 10-12 minutes per side, turning with tongs to develop an even, charred crust and ensuring the juices remain clear.
Check for a deep golden-brown, crispy skin and internal temperature of 165°F (74°C) to ensure doneness.
Remove the drumsticks from the grill and let them rest for 5 minutes; this allows juices to redistribute and results in tender, juicy meat.
Serve the grilled drumsticks hot, with a smoky aroma, crispy skin, and tender meat ready to enjoy.
