I've been eating lobster my whole life here in Boston and for most of it, I did exactly as Mike said. However I have a bunch of pickling spice on hand and there's nothing wrong with making a court bouillon with some pickling spice maybe a few onions and a lemon.
@rpooley what's the presentation on lobster? I guess my answer is based on that. If you serve it whole or even cut it up a bit but they are going to pick at it, the court bouillon aromas will come through cause it's on the shell. I'd argue it's not so sexy, but it is easy for back of house. If you're going to pick out the meat and do something with it like mac and cheese or bisque or something, I'd just do what Mike said. The spices wouldn't come through in that case.
If it was me, I'd steam it like 80% done, chop it in half and then finish on the grill with a compound butter. Broil too long it could get tough, you just want that last bit for the flavor of browning