For the lobster jus: Cut the lobster into quarters. Set aside into a draining board.
Bring a soup pot to high heat with olive oil. Once hot, add the lobster quarters. Saute the lobster for about 5 minutes, then strain the excess fat from the pot, then flame the lobster with the brandy. Add the carrot, onion, fennel and garlic, stir for about 8 to 10 minutes on medium heat, then add the tomato paste, stir for another 5 minutes then add the ice. When melted down, it should cover all the ingredients of the soup. If not, just add some cold water until covered. Bring the soup to a boil, then simmer for 1 hour and 45 minutes.
Remove the lobster claws and keep chilled. With the help of a heavy-duty hand blender, crush and blend the soup with the remaining components. Blend until everything is well crushed and almost smooth.
With the use a chinois, or fine mesh sieve, strain the soup with the help of a ladle, by pressing down through the chinois, to extract all juices from lobster and the other components.
In a clean and dry heavy soup pot, pour the lobster jus and reduce it by half. When reduced, add the butter, (very chilled and diced into cubes), gradually puree with a hand blender until smooth. Return the soup pot to a simmer for 5 minutes. Then set aside.
For the truffle foam:
Chopped the truffle and the shallot very fine, then whip the heavy cream until smooth, almost runny.
Add the truffle, shallot, peppercorn and salt to taste. If desire for more truffle flavor, you can add a drop of white truffle oil. Keep aside to cool.
Take the coral from inside the carcass of the lobster, very bright velvet green color membrane.
With a chef knife, chopped very finely, insert the coral on a sheet pan with a wax paper, then bake it at 225 degrees for about 20 minutes or until it turns red. Then cool it in a dry environment. When cool, with a coffee grinder, bring to a chunky powder.
Add to fine Chopped lemon thyme.
Remove lobster meat from the chilled claws, discard the shells. Keep lobster meat cool until serving.
Pour the hot lobster jus into 3/4 cappuccino cup add 1 tablespoon of the truffle cream and a sprinkle of garnish, per cup.