Melt the butter in a microwave-safe dish. Stir in lemon juice, chives, garlic, hot sauce, and pepper. Reserve half of the garlic butter for dipping in a separate bowl.
Using kitchen shears, butterfly the lobster trails down the middle of the softer underside of the shell. Cut the meat down the center without cutting all the way through. Inset a metal skewers so that the lobster tail stands straight. (You can also just cut the lobster tail down the middle if you want all the way through, making two halves).
Brush the lobster tails with some of the garlic butter. Grill the lobster tails over medium high heat for about 5 minutes or until the shells are bright in color. Brush with the garlic butter again. Continue to grill for about 4 minutes or until the lobster meat is opaque in color.