Bring the 8 cups water to a boil in a large pot. Add the lobsters and bring the water back to a boil, then cover, reduce the heat, and simmer the lobsters gently for 5 minutes. Leave them in the cooling liquid for 10 minutes off the heat, then lift them from the liquid and set them aside until cool enough to handle; set the pot aside.
Crack the shells and remove the meat and any red coral, from the shells. Reserve the shells. Cut the meat of each tail lengthwise in half, and remove and discard any remaining intestinal tract. Set the coral aside.
Put the shells back in the pot with the cooking liquid, bring to a boil, and boil, covered, for 30 minutes. Strain the stock. (You will have about 7 cups.) Set 1 cup of stock aside in a small saucepan for the sauce and freeze the remainder for use in a bisque or for making a soup with pasta.
FOR THE ARTICHOKE HEARTS: Preheat the oven to 350 degrees.
Put the artichoke hearts in a saucepan, add 1 cup water, the oil, lemon juice, and salt and bring to a boil. Cover, reduce the heat, and boil the artichokes gently for about 20 minutes, until they are tender and most of the cooking liquid has evaporated. Let cool.
When the artichokes are cool enough to handle, remove and discard the chokes.
At serving time, reheat the artichoke hearts for about 1 minute in a microwave oven, or for 10 minutes in a 200-degree oven; set aside in a warm spot. Put the red coral roe if you have it, in an ovenproof dish and heat it for 5 minutes in the oven to dry it. Coarsely chop the coral.
Just before serving, reheat the lobster meat for 20 to 30 seconds in the microwave oven or for 8 to 10 minutes in a 200-degree oven.
Meanwhile, bring the cup of reserved stock to a boil and boil until it is reduced to 1/2 cup. Add the butter, chives, and salt and pepper to taste and bring to a strong boil.
Place an artichoke heart on each of four plates and arrange the meat of half a lobster tail in the cavity of each, with a piece of claw meat alongside. Spoon the sauce on top, sprinkle with the roe, and serve.
HOW TO TRIM AN ARTICHOKE
With a sharp knife, trim 1 1/2 inches from the top of each artichoke. Using scissors, trim off the top thirds of the remaining leaves to remove the sharp "prickers." Trim the artichoke stems and peel the fibrous outer surface from the stems.
Copyright © 2011 by Jacques Pépin. Used by permission of Houghton Mifflin Harcourt. All rights reserved.
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