My mom grilled lobster a couple of times when I was a kid. I can’t remember if they were split longitudinally, or if they were just the tails. I assume they were basted with butter.
So. How do you grill a lobster? Aside from putting a bright light on it and asking, ‘Where were you on the night of April 15th?’
I use lemon juice, olive oil, garlic powder, white pepper, paprika and salt. I marinade the tails in that and then baste 'em when they’re on the grill. Alternately (and more easily), I drench them in butter and keep on applying it when they’re being grilled.
Dunno about anybody else, but I salt-grill the tails. Big pile of rock salt in a baking dish, heat it up, scoop out slots in the hot salt (use something heat-resistant), stick in the tails (with a probe thermometer inserted), cover up with the scooped-away salt, return to the oven, wait for the thermometer to beep, done. Neat trick.
I’ve grilled 'em whole a couple times. It’s a little heavy on the ick though. I par boil or steam them about half way done. I then yank the critters from the pot & put them in an ice bath, until they’re cool enough to be handled. Now for the ick…
Next, I split them all the way from the eyes to the tail and scoop out all the organs and other gross stuff in the cavity. I’ll then brush any exposed meat with butter and apply a healthy dose of cajun seasoning. Then they go on the grill for the final cooking.
I suppose you could do the whole thing on the grill. I prefer to have the boiling water play the role of executioner though.
I just buy and grill the tails. The tails we get are pretty large, so one (maybe one and a half) feeds one of us; along with a good side dish, it makes a tasty meal.