Preheat the oven to 375 degrees F. Line a baking sheet with aluminum foil and set aside.
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.
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.
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.
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.
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.
Place 1 lobster half on each of 4 large plates, garnish with additional parsley, and serve immediately.
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.
*Bouquet Garni is a small bag filled with herbs and seasonings that is used to flavor poaching liquids and broths. By placing the items in a bag, the liquid stays free of debris and the seasonings can easily be extracted. The ingredients in the bouquet garni vary to suit the final dish.
Recipe courtesy of Emeril Lagasse, originally appearing in Emeril's Delmonico: A Restaurant with a Past, William Morrow Publishers, New York, 2005
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