Hold lobster with tail away from you. Cut on the right side of shell towards the tail. Repeat the process on the left side. Then make a cut across the back to free the top of the shell.
Rinse the tail with cold water to remove vein and any bits of shell. Pat dry with paper towels. Release meat from shell with the tail end still attached. Place the meat on top of the shell.
Wrap tails with small pieces of aluminum foil to prevent burning.
Combine melted butter, garlic, thyme, pepper, and salt in a small bowl. Brush marinade over lobster flesh.
Place lobster tails on a preheated grill. Grill for 5 minutes, and then baste with remaining garlic thyme butter. Continue to grill for another 5 minutes.
Remove and serve immediately.