STEP 1) In a large enameled or stainless steel pot, simmer the white wine, onion, carrot, celery, parsley, bay leaf, thyme, peppercorn, and tarragon for 15 minutes. (wait to add the lobsters!)
Bring to a rolling boil and then add the live lobsters. Once the lobsters are cooked (you can tell when they are done once they turn bright red and the long head-feelers can be pulled out pretty easily), remove them from the pot.
STEP 2) While the lobsters are steaming, stew the mushrooms over low to medium low heat in a covered saucepan with butter, lemon juice, and salt for 10 minutes.
Pour the mushroom juices into the lobster-juices pot (not the mushrooms! set aside for end of recipe) and boil down rapidly until it is reduced to about 2 1/4 cups liquid.
Strain the lobster-mushroom liquid into a large saucepan and bring to a simmer.
STEP 3) In another saucepan, cook the butter and flour together slowly. Be sure not to brown it!
Remove from heat, and beat in the lobster-mushroom liquid.
Boil combined liquid, stirring constantly for 1 minute. Drizzle the 1 tablespoon cream on top of the sauce. Remove from heat.
Split the lobsters in half, lengthwise, making sure to keep the shell halves intact. Discard 'sand sacks' in the heads and intestines.
STEP 4) Thin out the sauce with additional whipping cream (by now the sauce should be thick enough to coat a spoon pretty heavily).
Set aside, top filmed with a spoonful of cream.
Remove the meat from the lobster tails and claws, cut into 3/8 inch cubes.
STEP 5) Set another skillet with 4 tablespoons butter over medium heat.
When butter foam begins to dissipate, stir in the lobster meat and saute, stirring slowly, for 5 minutes, until the meat has turned a rosy color.
Pour the cognac in and boil for a minute or two, shaking the skillet, until the liquid has been reduced by half.
STEP 6) Preheat oven to 425°F.
Fold cooked mushrooms and 2/3 of the sauce into the skillet with the lobster meat.
Arrange the split lobster shells on a shallow roasting pan. Heap the lobster mixture into the shells, and cover with the remaining sauce.
Sprinkle with cheese and dot with butter (you can refrigerate it at this point to finish up later on).
Place in upper 1/3 of the oven for 10-15 minutes, until lobster is bubbling and sauce is a nice brown. Serve immediately!
Recipe by Food, adapted from Julia Child.
