Prepare lobster. Boil the lobster tails in salted water for about 5-7 minutes until cooked through. Once cool, remove the lobster meat, chop it into bite-sized pieces, and set aside. Keep the shells for later.
Make the stock. Heat the olive oil in a large pot over medium heat. Add the lobster shells, onion, carrot, celery, and garlic. Sauté for 5-7 minutes until vegetables soften and the shells are fragrant.
Add tomato paste and liquids. Stir in the tomato paste, then add the brandy or cognac. Allow it to cook for a minute to let the alcohol evaporate. Add white wine and simmer for another 3 minutes.
Simmer the soup. Add the seafood stock, bay leaf, thyme, and some salt and pepper. Bring the soup to a boil, then reduce heat and simmer for about 30 minutes to develop the flavors.
Blend and strain. Remove the lobster shells and bay leaf. Blend the soup using an immersion blender (or transfer it to a blender in batches). Strain the soup through a fine mesh sieve to get a smooth, silky texture.
Finish with cream and butter. Return the strained bisque to the pot, stir in the heavy cream, and let it gently simmer for another 5 minutes. Add butter for extra richness.
Add lobster meat. Stir in the chopped lobster meat and heat through for 2-3 minutes.
Serve. Ladle the bisque into bowls, garnish with parsley or chives, and enjoy!
