If you are serving the shellfish with mayonnaise, make it as usual and stir in crème fraîche and a little lemon juice.
Heat the overhead grill. Mix the butter with the garlic and parsley and a little salt and pepper. Split the live lobsters or crawfish in half lengthwise with a very large, sharp knife, starting at the middle of the head shell. Remove the intestinal veins. Crack the claws: try a few blows with a mallet, but not hard enough to crush them completely, or use the back of a cleaver.
Line a roasting tin with foil, put in the lobsters, cut side up, and spread them with the garlic butter. Grill for 10 minutes and serve immediately a plain green salad is all you need to partner it.