1 Separate lobster tails and claws. Cut tails in half lengthwise with a heavy knife and remove the intestines.
2 Remove the meat from the lobster shell and cut into 1 cm (approximately 1/2-inch) pieces.
3 Cut open claws with a knife.
4 Use knife to cut the shell off meat. Cut legs in half horizontally.
5 Rinse meat under cold running water, cover and chill. Chop the shell coarsely.
6 Remove dark green part of leek, cut a 3 cm (approximately 1-inch) long piece of the white part and set it aside.
7 Trim fennel and remaining white part of leek and cut into 2 cm (approximately 1-inch) pieces. Peel garlic and cut into thin slices.
8 Heat oil in a large pot. Fry shells over medium heat for 4-5 minutes.
9 Sauté shells, sliced leek, fennel, garlic, star anise and whole chile pepper for 1 minute. Add cognac. Add white wine and fish stock and bring to a boil. Add salt and peppercorns. Cook for about 1 1/2 hours over medium heat.
10 Meanwhile, put cherry tomatoes into boiling water, remove, rinse in cold water and remove skin.
11 Rinse reserved leek portion and cut into quarters, then cut into 1 cm (approximately 1/4-inch) pieces. Cook for 30 seconds in boiling salted water, remove, rinse in cold water and drain well.
12 Pour broth through a cheesecloth-lined sieve into another pot.
13 Heat leeks, tomatoes and lobster in about 200 ml (approximately 3/4 cup) of broth. Take out with a slotted spoon and place in preheated soup plates. Fill with very hot lobster broth and serve the lobster soup immediately.