Simmer all of the ingredients under Step 1 (except the lobsters!) in a large enameled or stainless steel pot for 15 minutes.
Bring to a rolling boil and then add the live lobsters.
You can tell when they are done because they are bright red and the long head-feelers can be pulled out pretty easily.
While the lobsters are steaming, stew the mushrooms over low to medium low heat in a covered saucepan with all of the other ingredients from Step 2 for 10 minutes.
When the lobsters are cooked, remove them from the kettle.
Pour the mushroom juices in the kettle (not the mushrooms!) lobster-juices pot and boil down rapidly until it is reduced to about 2 1/4 cups liquid.
Strain the liquid into a large saucepan and bring to a simmer.
Cook the first 2 ingredients from Step 3 slowly together in another saucepan.
Be sure not to brown it!
Remove from heat, and beat in the lobster-mushroom liquid.
Boil, stirring constantly, for 1 minute.
Drizzle the cream from step 3 on top of the sauce.
Split the lobsters in half, lengthwise, making sure to keep the shell halves intact.
Discard 'sand sacks' in the heads and intestines.
Rub 'lobster coral' and 'green matter' through a fine sieve into a 3 quart mixing bowl, and blend it into the ingredients from step 4.
Beat the lobster-mushroom sauce into this mixture in small spats.
Return the sauce to the pan, and, stirring with a wooden spoon, bring it to a boil.
Boil slowly for 2 minutes.
Thin out with step 5.
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.
Set the skillet with the butter from step 6 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.
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 or fireproof serving platter.
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! ?