Heat the olive oil in a dutch oven or large heavy-bottom saucepan over low heat. Add the leeks, shallot and celery with a pinch of salt. Allow the vegetables to sweat for several minutes, stirring periodically, until softened. Add the butternut squash and cook for several more minutes, stirring.
Turn the heat up to medium and allow a brown glaze to slowly build on the bottom of the pan by stirring less often. Keep an eye on it to make sure it doesn't burn. Use a spatula to scrape up and incorporate the brown bits a few times. After several minutes, add 1/3 cup cognac to deglaze the pan and loosen the rest of the brown bits. Allow to alcohol to reduce down to almost nothing, stirring the vegetables frequently, and then add the fish stock and thyme leaves.
Bring the liquid to a boil and then cover the pot and reduce the heat to low. Allow the soup to simmer for 45 minutes.
Carefully transfer the soup to a blender and puree until very smooth. Wipe down the dutch oven and return the soup to the pot. Stir in the half and half, 1 tablespoon cognac, and the salt, pepper and cayenne. Add the lobster meat, optionally reserving some for garnish. Serve hot. To reheat the soup, strain out the lobster meat first to avoid overcooking it. Add it back in once the soup is hot.