For the lobsters: Place the lobsters into a large stainless-steel pot and add about 4 cups of water. Cover, and bring to a strong boil, which may take nearly 10 minutes. Reduce the heat to medium, and continue boiling gently, still covered, for about 5 minutes. Set aside, covered, for 30 to 45 minutes, until the lobsters in the pot are cool enough to handle. Retain the broth for use in the sauce. Break off the tails and claws from each of the lobsters, and crack the shells, taking care to retain any liquid released and add it to the broth. Remove all the meat, split the tail in half lengthwise, and remove and discard the intestinal tract. Cut each half tail into 3 or 4 pieces, and halve the claw meat. Arrange all the meat in a gratin dish lined with plastic wrap. Set aside. The recipe can be prepared ahead to this point and refrigerated. Reserve the shells for the sauce.
For the Americaine sauce: Heat the olive oil in a very large saucepan. Cut the lobster bodies open, and remove and reserve any liquid tomalley or row in a bowl. Cut each body into 4 or 5 pieces and add to the pan along with the shells from the tails and claws. Saute over high heat for 10 to 12 minutes, until the shells begin to brown and become crusty, and most of the moisture has evaporated. Add 2 tablespoons cognac to the pan and carefully light it to flambe the shells. Add the onion, carrot, and celery and cook with the shells for 4 to 5 minutes. Add the tomato, thyme, tarragon, bay leaves, tomato paste, wine, and reserved lobster broth. Bring to a boil and boil gently for 40 to 45 minutes. Season with salt and pepper, to taste.
Strain the mixture through a colander set over a bowl, and shake the colander to get out as much juice as possible. Strain the juice again through a very fine strainer. Pour the liquid into a saucepan and reduce it to about 3 cups. Mix the potato starch (or arrowroot or cornstarch) with the water and stir it into the boiling reduced liquid. Return it to a boil and stir in 1/2 cup heavy cream, remaining 2 teaspoons cognac, and salt and pepper to taste. The sauce is now ready. Break cold butter into pieces and distribute them on top of the sauce. When the butter has melted, spread it lightly over the surface of the sauce with the tines of a fork. This will prevent the sauce from forming a skin or discoloring. The butter will be stirred into the sauce at serving time.
Prepare the lobster:
When you are ready to complete the recipe, melt 4 tablespoons butter. Unwrap the plastic wrap from the reserved lobster and pour the butter gently over the meat. Season with salt and pepper and sprinkle on the tarragon. Bring the plastic wrap back over the lobster meat so the meat is held tightly together and can soak up the butter.
For the souffle: Make the bechamel sauce by melting 3 tablespoons butter in a saucepan and cooking until it is light brown in color to give some intensity to the sauce. Add the flour and mix well. Stir in the milk and season with salt and pepper. Mix well with a whisk, bring to a boil, and boil for about 30 seconds. Continue whisking while adding 4 egg yolks to the hot bechamel and stir well to incorporate. (Reserve the remaining 2 egg yolks for another recipe.)
Preheat the oven to 375 degrees F. Lightly butter a 6 or 7-cup gratin dish and sprinkle with 2 to 3 tablespoon of the grated Parmesan, tilting the dish so the cheese sticks to the butter and coats the bottom and sides. Tap out and reserve any excess cheese for use on top of the souffle. Beat the 6 egg whites until firm but not dry. Add about 1/3 of the egg whites to the bechamel and mix with a whisk to incorporate. Add the remaining whites and the Gruyere and fold in with a rubber spatula. It is important to work quickly so the beaten egg white gets incorporated into the mixture without getting grainy. Pour into the prepared dish and sprinkle on the remaining Parmesan. Place in the center of the oven and bake for about 35 minutes, or until golden, crusty, and brown.
While the souffle bakes, place the lobster meat into a 170 degree F oven for 20 to 25 minutes, or just until warmed through, leaving it in the plastic wrap. The lobster should be warm but not hot. If heated too fast or at too high a temperature, it toughens. Just before serving, heat 6 soup plates in the oven.
Divide the lobster meat among the warm soup plates, bring the sauce to a boil, and spoon it over the lobster meat in each plate. Spread the meat to create a little space in the center of each plate. Spoon about 1 cup of the souffle, along with some of the crust, into the middle of each plate. Garnish with truffle slices, if using.