Cut the onion, leek, carrot, celery and garlic into small pieces
Heat the olive oil in a pan and in it first fry the onions and garlic until glazy, then add the sliced leeks, carrot and celery. Stir well until the flavors loosen, remove the pan from the heat and set aside.
Put olive oil in a second pan and fry the lobster shells all around, add salt, pepper and cayenne pepper. Set the shells aside and fry the tomato puree to de-acidify it, stir well and deglaze with a splash of cognac.
Add the sautéed vegetables and stir well. Then add the broth and wait until it comes to a gentle boil. Put the lid on the pan and simmer until the broth is intense. This takes about 1.5 hours. I put the pan in the oven at about 140 degrees and it will cook itself.
Pour the liquid through a sieve. To bind the soup a little, make a small roux of equal parts butter and flour to which you add the liquid little by little so that you get a nice lightly bound whole. Taste to see if the mixture needs additional pepper and/or salt, but due to the concentration of the liquid this is unlikely.
Preheat small cups, put in them the lobster cut into pieces that you have removed from the claws, and add the hot liquid.
Done is your lobster soup.