Fill a large stockpot with water and add about 2 tablespoons of salt per quart. Bring to a rolling boil over high heat.
Plunge the live lobsters head-first into the boiling water, cover, and cook for 8-10 minutes until shells turn bright red. Remove with tongs and cool on a cutting board for 10 minutes.
Twist off the claws and tails from the bodies. Use kitchen shears to cut down the center of each tail shell on the underside and carefully remove the tail meat in one piece.
Crack the claws with the back of a knife or nutcracker and extract all the meat. Remove the dark vein from each tail. Chop the lobster meat into bite-sized pieces, keeping a few larger chunks for presentation. Reserve the empty tail shells.
Melt 4 tablespoons of butter in a large skillet over medium heat. Add the minced shallots and cook for 2-3 minutes until translucent and fragrant.
Stir in the chopped mushrooms and cook for 5 minutes until they release their moisture and start to brown. Pour in the cognac and let it simmer for 1 minute to cook off the alcohol.
Sprinkle the flour over the mushroom mixture and stir constantly for 2 minutes to create a roux. Gradually whisk in the heavy cream and fish stock, stirring constantly to prevent lumps.
Add the Dijon mustard, chopped parsley, and tarragon. Season with salt and white pepper to taste. Simmer for 5-7 minutes until the sauce thickens enough to coat the back of a spoon.
Reduce heat to low and gently fold the chopped lobster meat into the sauce. Stir carefully to coat all the lobster pieces without breaking them apart. Let the mixture warm through for 2-3 minutes.
Preheat your broiler to high and position the oven rack 6 inches from the heat source. Arrange the reserved lobster tail shells on your baking sheet and spoon the lobster and sauce mixture generously into each shell.
Mix together the grated Gruyere and Parmesan cheeses. Sprinkle evenly over the top of each filled shell and dot with the remaining 2 tablespoons of butter.
Place the baking sheet under the broiler for 3-5 minutes, watching carefully, until the cheese is golden brown and bubbling. The edges should be lightly caramelized but not burned.
Transfer the broiled lobster shells to serving plates using a wide spatula. Garnish with fresh parsley and lemon wedges. Serve right away while the cheese is still bubbling.