Preparation time: 25 minutes
Cooking time: 55 minutes

1. Bring the water to a boil in a large saucepan. Add the onion, carrot, thyme and bay leaf. Reduce the heat to low and poach the lobster in this court-bouillon for 10 minutes. Remove the lobster and cool slightly. 
2. Shell the tail and claws, keeping them for other uses, since they won't be needed in this recipe. Keep the coral and the shell which will be used for the soup base.
3. Sauté the onion and tomatoes in a large saucepan. Once they begin to color, add the coral and the lobster shell, coarsely chopped, and sauté everything for 3 minutes over low heat. 
4. Add the cognac and white wine and reduce over low heat for 10 minutes. After the alcohol has evaporated somewhat, add the 15% cream and the milk. Continue cooking over low heat and reduce for 30 minutes. 
5. Pour into a blender, whir everything and strain to remove any pieces of shell. 

For an original presentation
Pour the soup into a coffee cup. A few drops of truffle oil sprinkled over the surface with a teaspoon will add an unexpected subtle flavor. For even more effect, top with a spoonful of whipped cream or some steamed milk foam.