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When the coals are heated and ready, split the lobster in half lengthwise down the center of its body, then take the back of your knife and crack the claws so they cook evenly with the tail, making sure you remove the rubber bands as well.
Grill the lobster cut-side down to cook very quickly and to achieve some char on the exposed meat, about 2 minutes at most, then flip so the shell is facing the flame. Pour some of the butter mixture into the cavity to poach the lobster meat, reserving some for serving. Poach the lobster until the meat has shriveled up a bit inside the shell and is a brilliant white, another 4 minutes.