Prepare the grill for two zone or direct/indirect cooking with lump charcoal. Target 450 - 500 degrees Fahrenheit over the direct heat.
While the grill heats up, combine the salt and pepper in a small bowl. Coat the steaks with olive oil and dust them with some of the salt and pepper. Save some of the seasoning for the lobster.
Prepare the lobster by placing 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 lobster tails with the remaining salt and pepper. (no olive oil needed)
Place the steaks over direct heat for 5 minutes and cover the grill. Flip the steak over, cover grill, and grill for another 5 minutes. Then move to the indirect side and cover. Continue grilling over indirect heat until the internal temperature reaches your desired doneness. We like 125 degrees F for rare (about 8 minutes). Remove and let rest for 10 minutes and place one teaspoon of the butter on each steak while resting (it will slowly start to melt into the meat).
While the steak is resting grill the lobster.
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 lobster meat and then serve warm with your favorite sides.