01 - Bring a large pot of salted water to a boil. Add live lobsters and cook 7–8 minutes until shells turn bright red. Transfer to ice water to cool. Remove meat from claws, tails, and knuckles; reserve shells. Chop meat into bite-sized pieces and refrigerate.
02 - Heat butter and olive oil in a large pot over medium heat. Add onion, carrot, celery, and garlic. Sauté 6–8 minutes until softened.
03 - Stir in tomato paste and cook 2 minutes. Add reserved lobster shells and cook 2 more minutes, stirring constantly.
04 - Pour in white wine, scraping browned bits from pot bottom. Simmer 2 minutes to reduce slightly.
05 - Add seafood stock, bay leaf, thyme, and cayenne. Bring to boil, then reduce heat and simmer uncovered 30 minutes to develop flavor.
06 - Remove and discard lobster shells and bay leaf. Use immersion blender to purée until completely smooth. Alternatively, transfer to blender in batches.
07 - Return bisque to pot. Stir in apple cider vinegar, heavy cream, and reserved lobster meat. Simmer gently 5 minutes. Season with salt and pepper. Ladle into bowls, garnish with chives or parsley, and serve hot.