Step 1Put the butter in a small mixing bowl and roughly beat in the minced shallots, parsley and lemon juice with a spatula or fork. Form the butter into a rough log about 5 inches long on a sheet of wax paper. Roll it tightly to form the log into a smooth cylinder and twist the ends to seal. Refrigerate until firm, at least 1 hour.
Step 2Peel the cucumbers, removing the tough green skin. Then continue peeling, shaving long thin strips like pasta noodles. It's easiest to do this with the cucumber resting on a cutting board, leaving one end of the cucumber unpeeled for a better grip. Keep rotating the cucumber so you're peeling around the seedy center without cutting into it. When you get to the seeds, stop. Discard the center with seeds. You should have about 4 cups of long strips about one-half-inch wide.
Step 3Heat the grill over a medium-hot fire. In a small saucepan, melt half of the shallot butter over low heat.
Step 4On a sturdy cutting board using a heavy, sharp chef's knife, kill and halve the lobsters lengthwise. First, drive the point of the knife down through the head of the lobster where the base of the head joins the body, bringing the knife all the way down to split the head in two lengthwise; this will kill the lobster. Continue cutting the lobster in half lengthwise. Remove the vein that runs along the top of the tail (as with shrimp), as well as the sac near the head. Using the back of a heavy chef's knife or the flat side of a kitchen mallet, give the claws a good crack.
Step 5Brush the lobster meat with the melted butter, place the halves meat-side down over the medium-hot fire. Cook just until flesh is marked and starting to firm, about 2 minutes.
Step 6Turn the lobsters over, cover the grill, and continue cooking until the meat is firm and starting to pull away from the shell, about 5 to 7 minutes. Brush occasionally with any remaining melted shallot butter. When the lobsters are done, remove from heat to a platter and keep in a warm place.
Step 7When almost ready to serve, sprinkle the cucumbers with the salt, dill and vinegar; toss to combine. Do not season the cucumbers in advance or they will wilt and become mushy.
Step 8Cut the remaining chilled butter into 6 coins and place one on top of each lobster tail. Serve immediately with the cucumber strips alongside.