Step 1

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

Step 2

Bring a pot of salted water containing the lemons, quartered onion, and bouquet garni to a boil. Add the lobsters and cook until red and firm, 8 to 10 minutes. Transfer the lobsters to an ice bath to stop the cooking process.

Step 3

When the lobsters are cool enough to handle, cut in half lengthwise with a heavy sharp knife and carefully extract the tail meat. Remove the large claws from the body and gently crack with the back of a heavy knife to extract the meat. Gently pull the front legs from the shell and discard. Dice the tail meat and claw meat and set aside.

Step 4

Place the halved lobster shells on a baking sheet, cut sides down, and roast until dry, 5 to 6 minutes. Let cool on the baking sheet.

Step 5

Melt the butter in a medium saucepan over medium heat. Add the shallots and garlic and cook, stirring, until fragrant, about 30 seconds. Add the flour and whisk to combine. Cook, stirring constantly with a heavy wooden spoon to make a light roux, about 2 minutes. Add the cognac and cook, stirring, for 10 seconds. Add the milk slowly, stirring constantly to incorporate. Bring to a boil, reduce the heat, and simmer until thick enough to coat the back of a spoon, 2 to 3 minutes. Slowly add the cream, stirring constantly, until all is incorporated. Cook, stirring, over medium heat for 1 minute. (The mixture will be very thick.) Add the salt and pepper and stir well.

Step 6

Remove from the heat and stir in 1/2 cup of the cheese, the 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 a portion of the remaining cheese and broil until the top is golden brown, 5 minutes.

Step 7

Place 1 lobster half on each of 4 large plates, garnish with additional parsley, and serve immediately.

Step 8

Place the bay leaf, thyme, parsley sprigs, and peppercorns in the center of a 6-inch square piece of cheesecloth or a large paper coffee filter. Draw up the sides to form a pouch and tie with kitchen twine or unflavored dental floss.