Prepare the grill for direct cooking targeting 500 degrees Fahrenheit on the grill.
Place the lobster tails on a cutting board with the hard shell facing up. Using a chef knife or kitchen shears, gently cut the shell starting about 1/2 inch from the tail and working your way to the side that is open with the meat. Take care not to cut all the way through the other side.
Gently open the shell, exposing the meat using two hands. You will hear some cracking. Remove any shell that may break from the meat.
Season the flesh of the tails with the salt and pepper.
Place the lobster tails over direct heat, meat side down for approximately 5 minutes or until it gently pulls away from the grill without sticking.
After flipping add about a 1 heaping teaspoon medallion of the compound butter over the meat. Grill for an additional 3 - 4 minutes over direct heat. The butter will slowly melt. When the internal temperature of the lobster meat reaches 135 degrees F it is done.
After removing from the grill, place another teaspoon medallion of butter on the meat and then serve warm with your favorite sides.