Tie the parsley, thyme, tarragon, and bay leaves with twine. Place in a large stock pot with garlic, wine, water, mushrooms, onion, carrot, celery, and peppercorns. Bring to a boil, then simmer partially covered for 30 minutes.

While the stock simmers, prepare the lobsters by laying them face to face and tying the tails together. Freeze the lobsters for 15 minutes to put them to sleep or use a knife to kill them quickly.

Prepare a bowl of ice water to chill the cooked lobsters.

Place the lobsters in the hot stock, bring to a boil, cover, and cook for ten minutes. Use tongs to transfer the lobsters to the ice water to stop the cooking. Cut the twine.

Once cool, place a cutting board inside a sheet pan to catch drips. Tear off the legs, add to the stock pot. Remove claws and knuckles, pick out the meat, and add empty shells to the stock. Boil and simmer until reduced to two cups. Strain and discard solids.

While the stock cooks, cut the lobsters in half lengthwise with kitchen shears and a sharp knife. Remove and discard the intestinal tract and innards. Cut the tail meat into bite-sized pieces. Rinse shell halves and place on a foil-lined tray.

Melt half the butter in a large skillet over medium heat. Add lobster meat and sauté briefly. Remove to a bowl with a slotted spoon.

Increase heat to medium-high, add mushrooms, and cook until browned. Transfer to the bowl with lobster.

Add remaining butter to the skillet, reduce heat to medium, and sauté shallots for two minutes. Turn off heat, pour in cognac to deglaze, then return heat to medium-low. Stir in flour to make a roux, cooking for two minutes.

Add one cup of stock, stirring to combine. Add Dijon, Parmesan, and cayenne. Stir in cream, lobster, and mushrooms. If too thick, add more stock. Taste and season with salt and white pepper if needed.

Remove from heat and divide filling among the shell halves. Top with Gruyere cheese and broil until golden brown.

Sprinkle with chopped parsley and serve.