Heat the olive oil in a 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 squash and cook for several 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. 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 the alcohol to reduce 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.
Transfer the soup to a blender and puree until very smooth. Wipe down the saucepan 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. Top with brown butter sage drizzle.