Prepare the lobster tails. Use kitchen scissors to cut along the top of the shell, down the middle, all the way to the tip of the tail.
Use your fingers to to gently pull the meat up through. Lay the meat on top of the shell.
Place the butter, lemon juice, Cajun seasoning and black pepper in a small bowl and heat in the microwave until melted.
Brush each lobster tail with the melted butter mixture.
Smoke the the lobster tails for about 1 hour or until the internal temperature reaches 140°F. Baste the lobster every 15 minutes with the remaining butter.
Notes
Use a mild wood for a subtle smoke flavor like alder, apple or cherry.
Baste the butter throughout the smoking process as it really flavors the lobster.
Use a meat thermometer so you know when the lobster is cooked.
If you're short on time, you can smoke at 300°F for about 15-20 minutes, but you'll have less smoke flavor.