To keep the lobster meat tender it is steamed for only 6 minutes, long enough to be able to remove the meat form the shells. The chopped lobster will finish cooking in the final reheat of the chowder.
In a 16- to 20-quart stockpot, bring 5 cups of water to a boil over high heat. Add 1 Tbs. salt and the lobsters, cover, and steam until bright red and one of the smaller legs twists off easily, about 6 minutes. Remove the lobsters with tongs. Leave the cooking liquid in the pot.
When the lobsters are cool enough to handle, twist off the claws and the tails. Using a nutcracker, crack the knuckles and claws and push out the meat with your little finger or a pick. Set the tail on a hard surface and use your hand to press down and crack the shell; push out the meat. Slice the tail meat in half lengthwise and remove the black intestinal vein. Coarsely chop the lobster meat. Put the lobster shells and any liquid from the lobsters back into the pot and bring to a boil. Boil for 15 minutes to flavor the broth and then strain through a large sieve into a bowl and reserve. You'll have about 4 cups.
In a 6-quart stockpot, melt the butter over medium heat. Add the shallots, fennel, and thyme and cook, stirring often, until the shallots and fennel are beginning to soften, about 3 minutes. Add the potatoes and cook, stirring, for 2 minutes. Add the reserved lobster broth and bring to a boil. Reduce the heat to low and simmer for 15 minutes or until the potatoes are just tender.
Stir in the lobster meat, Madeira, and half-and-half and bring back to a simmer. Remove the pot from the heat. Let the chowder sit for 5 to 10 minutes for the flavors to meld.
Season to taste with salt and pepper. Ladle the chowder into shallow bowls and garnish with chives.
Recipe adapted from Moveable Feast with Fine Cooking.