Plunge the live lobster in a large quantity of generously salted boiling water, cook for 8 to 10 minutes. Leave to cool in the cooking liquid. Break the stems off the artichokes, then using a sharp knife, peel by turning them to save only the base. Remove the choke with a sharp spoon, sprinkling lemon juice on the clean hearts, then cook in boiling salted water for about 10 minutes. Drain and rinse under cold water and leave to cool. Peel the lettuces, saving only the cores, separate the leaves and set aside. Shell the lobsters, slice the tails, brake the claws retrieving all the meat. Lay the lettuce heart leaves in individual serving dishes, thinly slice the artichoke hearts and place over the lettuce. Cover with nice pieces of lobster. Blend the mayonnaise with the tomato coulis in a mixing bowl, add the juice of 1/2 a lemon, the paprika and the Cognac. Stir and serve the sauce on the side.
