Place lobster on cutting board. Using a sharp chef’s knife held vertically, plunge the point into the lobster’s head about 1 inch behind the eyes. Push the knife completely in to touch the cutting board and then move it forward to cut the entire head in half. This is the quickest and easiest method of killing a live lobster.
Pull the claws from the body. Prepare an ice water bath in a bowl large enough to hold all the lobster parts and set aside.
Place the claws and bodies in the top half of a steamer over boiling water. Cover and steam the lobster for 4 minutes. Remove the bodies and continue steaming the claws for an additional 3 minutes.
Immerse both the bodies and the claws in ice water bath as soon as you remove them from the steam to stop the cooking.
Crack the shells on the bodies and claws, and carefully remove the meat, keeping it in pieces as large as possible. Separately reserve the meat and the shells.
Preheat over to 350°F. Place the lobster shells in a roasting pan in the preheated oven and roast, turning occasionally, for about 12 minutes or until nicely colored and fragrant. Remove from the oven and set aside.
Heat 2 tbsp. of the butter in a large saucepan over medium heat. Add the carrot, onion and celery, and sauté for about 4 minutes or just until the vegetables begin to soften without taking on any color.
Add the tomato paste and sauté for about 1 minute or just until well-incorporated. Stir in the reserved lobster shells, followed by ¼ cup of the brandy and 1 tsp. Tabasco. Cook for about 3 minutes, stirring to deglaze the pan.
Add the cream, stir to blend, and raise the heat. Bring to a simmer and then immediately lower the heat to a gentle simmer. Season with salt and pepper, and cook gently for about 1½ hours, or until very thick and well seasoned.
. Remove the sauce from the heat and pour it though a fine-mesh sieve into a clean container, pressing on the solids to extract all the flavor. Discard the solids and set the sauce side.
. Heat the remaining tbsp. of butter in a medium sauté pan over medium-low heat. Add the shallots and season with nutmeg. Cook, stirring constantly for about 2 minutes, or until the seasonings have colored and are fragrant. Add the reserved lobster meat and sauté for 1 minute. Add the remaining tbsp. of brandy, stirring to deglaze the pan. Add the reserved cream sauce, raise the heat, and bring to gentle simmer.
. Remove from the heat, and using a slotted spoon, transfer an equal portion of the lobster meat to each of 4 shallow soup bowls. Whisk a bit of the sauce into the egg yolk to temper it and then whisk the egg mixture into the sauce.
. Add lemon juice and Tabasco to taste, and if necessary, adjust the seasoning with salt and pepper. Pour the sauce over the lobster in each bowl. If using, spoon an equal portion of caviar into the center of each bowl and garnish with Brioche Batons. Serve immediately.<