Preheat the oven to 375˚ F. Line a baking sheet with aluminum foil and set aside.

Cut the lobsters in half lengthwise with a sharp knife and remove the tail meat.

Twist off the claws from the body and gently crack with the back of a heavy knife to remove the meat. Gently pull the front legs from the shell and discard.

Chop the tail meat and claw meat into bite sized pieces and set aside.

Place the halved lobster shells on the baking sheet and set aside.

Melt the butter in a deep skillet over medium heat. Add the shallots and garlic, stirring, until fragrant, about 30 seconds. Add the flour and whisk to combine.

Cook the flour mixture, stirring constantly to make a light roux, approximately 2 minutes.

Add the cognac and cook for 10 seconds, stirring constantly.

Slowly add the milk, stirring constantly to combine. Bring to a boil, reduce the heat, and simmer until thick enough to coat the back of a spoon, approximately 2 to 3 minutes.

Slowly add the cream, stirring constantly, until thoroughly combined. Continue cooking while stirring over medium heat for 1 minute. (The mixture will be very thick.) Season with salt and pepper.

Remove from the heat and stir in the parmesan cheese, mustard, tarragon, and parsley. Fold in the lobster meat.

Divide the mixture among the lobster shells and place stuffed side up on a clean baking sheet.

Sprinkle the top of each lobster with the gruyere and broil until the top is golden brown, 5 minutes.

Place 1 lobster half on each plate, garnish with additional parsley, and serve immediately.