1. Prepare the Grill for Medium Heat: Start by preheating your grill to a medium heat setting, around 375-400 F (190-200 C). Make sure the grates are clean to prevent sticking. This initial step is crucial for even cooking and achieving those desirable grill marks.

2. Combine the Garlic Butter Sauce: In a medium bowl, melt the unsalted butter. Add the minced garlic, dried oregano, salt, black pepper, and paprika (if using). Stir everything together until well combined. The mixture should be fragrant and smooth.

3. Coat the Chicken Drumsticks: Place the chicken drumsticks in a large bowl or a shallow dish. Pour about half of the garlic butter mixture over the drumsticks. Use tongs or your hands to toss the drumsticks, ensuring each piece is thoroughly coated. This initial coating will infuse the chicken with flavor as it cooks.

4. Grill the Drumsticks: Carefully place the coated drumsticks on the preheated grill. Arrange them in a single layer, leaving a little space between each piece for even cooking. Close the grill lid.

5. Cook and Baste for Tenderness: Grill the drumsticks for 10-12 minutes, then flip them over. At this point, brush the drumsticks generously with the remaining garlic butter sauce. Continue grilling, flipping and basting every 5-7 minutes, until the chicken is cooked through. You’ll know they’re done when the internal temperature reaches 165 F (74 C) with an instant-read thermometer inserted into the thickest part of the drumstick, avoiding the bone. The skin should be nicely browned and slightly crispy.

6. Rest and Serve: Once cooked, remove the drumsticks from the grill and transfer them to a clean platter. Loosely tent them with aluminum foil and let them rest for 5 minutes. This resting period allows the juices to redistribute, ensuring the chicken remains tender and moist. Garnish with fresh chopped parsley, if desired, before serving.