Examine your lobster tails to ensure they're already cut down the center with the meat lifted and sitting on top of the shells as specified. Pat the meat dry with paper towels—this helps it cook evenly and develop better color on the grill. Mince your garlic finely, zest and juice your lemon, chop the chives into thin pieces, and measure out all your seasonings into small bowls. Having everything prepped and ready ensures you can focus entirely on grilling once the heat is high.
While heating your grill to 450-500°F, lightly season the exposed lobster meat on each tail with about 1/4 teaspoon of kosher salt and a few grinds of black pepper—don't oversalt yet since the butter will add more seasoning. Allow the grill to reach full temperature (this takes 10-15 minutes), and lightly oil the grates to prevent sticking. I find that properly preheating is essential for getting a nice sear on the meat without it overcooking.
While the grill finishes heating, melt the butter in a small saucepan over medium heat, then add the minced garlic and cook for about 1 minute until fragrant—don't let it brown. Remove from heat and stir in the lemon zest, lemon juice, a pinch of cayenne pepper, and a light grinding of black pepper. Taste and adjust seasoning if needed. Keep this butter warm but off direct heat so the flavors infuse while you grill.
Place the lobster tails on the hot grill with the meat side facing up. Grill for 4-6 minutes without moving them—this allows the meat to develop a light golden color and cook through the first layer. You should see the exterior begin to turn opaque and slightly firm to the touch. Brush generously with the garlic butter from Step 3 about halfway through this cooking time.
Carefully flip the lobster tails so the meat side is now facing down on the grill grates for 4-6 minutes until the internal temperature reaches 135-140°F when checked with a meat thermometer in the thickest part of the meat. Brush the exposed shell side generously with remaining garlic butter. I like to use a meat thermometer here because it takes the guesswork out—you want the meat just cooked through but still tender, not rubbery.
Transfer the grilled lobster tails to serving plates, meat side up. Drizzle any remaining garlic butter over the top and sprinkle generously with the chopped fresh chives for color and a delicate onion flavor. Serve immediately with the lemon wedges on the side for squeezing over the meat just before eating.