Preheat your Big Green Egg to 250°F using indirect heat. Install the convEGGtor and stabilize temperature before cooking. Add 1–2 small chunks of light smoking wood.
Remove lobster tails from packaging and pat dry. Trim off the small swimmer legs from the underside for cleaner presentation.
Butterfly the lobster tails. Using kitchen shears, cut down the center of the top shell to the tail fin. Gently loosen the meat, keeping the end attached, and rest the meat on top of the split shell.
Lightly brush the exposed lobster meat with olive oil. Season evenly with BBQ rub or the DIY lobster seasoning.
Place lobster tails shell side down on the grate. Insert a thermometer probe into the thickest part of the meat if using.
Smoke at 250°F for about 30 minutes before checking temperature. Continue cooking until the internal temperature reaches 135–140°F. Total cook time is typically 30–45 minutes depending on size.
During the final 20–30 minutes of cooking, place cubed butter in a small cast iron pan and set inside the grill to melt and lightly absorb smoke.
Remove lobster tails from the grill at 135–140°F. Rest for 5 minutes.
Spoon melted smoked butter over the lobster meat. Serve immediately with lemon wedges and chopped parsley if desired.