Set all grill burners to high heat. Pat every piece of chicken completely dry using paper towels. Trim away any excess fat with a sharp knife. Season all sides generously with salt and pepper.

Turn off one burner and lightly grease the rack directly above it. Place the drumsticks over the turned-off side to cook with indirect heat. Close the lid and grill for 20 minutes. Flip each piece, then grill another 20 minutes or until a meat thermometer reads 165 degrees F at the thickest part.

While the drumsticks finish cooking, combine lemon juice, chopped tarragon, minced garlic, salt, and pepper in a large bowl. Slowly drizzle in the olive oil while whisking constantly so the sauce comes together into a smooth emulsified coating.

For a charred golden finish, turn the burner directly under the drumsticks to high heat and grill for 2 to 3 minutes per side until the skin crisps and browns. Transfer the drumsticks into the bowl with the sauce and turn to coat. Cover loosely with foil to keep warm while you cook the thighs.

Repeat the indirect heat process for the chicken thighs. Grill approximately 15 minutes per side until they reach 165 degrees F internally. Turn the burner to high for the final 2 minutes to get that same golden char. Add the thighs to the sauce bowl, turn to coat, and serve hot.