Do the mise en place, i.e. peel and chop the vegetables and arrange the other ingredients so that they are ready to use when you need them. Prepare a large pot of boiling water.
Rinse the lobster under cold running water, ensuring that it is clean from any slime. Hold the lobster firmly on a chopping board and insert a knife in it carapace (an inch or so behind its eyes). This will kill the lobster instantly.
Plunge the lobster in the boiling water and let it cook for 1 minute. Carefully take it out of the pot and put on a chopping board. Separate the head and the tail. Cut the tail into 4 or 5 rings. Separate the claws from the head and cut them across the articulation. Use the back of a large knife to crack the large claws. Do not peel the claws just yet; simply cracking them will ensure they release their flavour into the sauce when cooked, without the meat breaking up into small pieces while stirring.
Split the head of the lobster lengthwise. Remove the gills, i.e. the feathery objects located above the feelers. Scrape out the tomalley (greenish substance towards the end of the head) and reserve in a bowl.
Place a large pot on medium-high heat. Add the neutral oil and let it heat up properly. Season the lobster pieces with salt and pepper and add them to the pot. Sauté the lobster for a few minutes until the shell turns bright red and there is caramelisation at the bottom of the pot. Remove the lobster from the pot and reserve in a plate.
Lower the heat slightly and add the onion, carrots and celery to the pot. Sauté them for a few minutes until they soften. Add the garlic, cayenne pepper and the lobster pieces and continue cooking for another 2 minute.
Pour in the Cognac and deglaze the pan. If possible, ignite the Cognac to burn off the alcohol quicker. Let the Cognac reduce almost completely, then add the wine and the fish stock. Deglaze the pan if there are still caramelised bits on the bottom.
Add the tomatoes and the bouqet garni to the pot and bring it to a simmer. Adjust the heat so that the sauce simmers gently. Cook for 15 minutes.
Remove the parts of the lobster that contain the meat from the pot. Continue simmering the sauce for another 45 minutes. I like to pick the meat from the claws and the tail pieces and return the shells into the sauce so they can continue to release flavour.
If any foam or scum floats to the top, skim it with a ladle or a spoon.
Pass the sauce through a fine sieve and collect the sauce into a clean pot. Discard the solids and return the strained sauce to a medium heat and bring to a simmer again.
Mash the tomalley with the softened butter and the flour until they form a smooth paste. Once the sauce is simmering, add small amounts of the paste while stirring with a whisk. The sauce will thicken because of the flour. Stop adding the paste when the desired consistency is achieved.
Add the cream and stir. Season with salt and pepper to taste. Serve.