01 - Bring a large pot of salted water to a boil. Add live lobsters and cook for 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 - In a large pot, heat butter and olive oil over medium heat. Add onion, carrot, celery, and garlic; sauté for 6–8 minutes until softened.
03 - Stir in tomato paste and cook for 2 minutes. Add reserved lobster shells and stir for another 2 minutes to develop flavor.
04 - Pour in white wine, scraping any browned bits from the bottom. Simmer for 2 minutes to reduce alcohol.
05 - Add seafood stock, bay leaf, thyme, and cayenne. Bring to a boil, then reduce heat and simmer uncovered for 30 minutes.
06 - Remove and discard lobster shells and bay leaf. Use an immersion blender to purée until smooth, or transfer to a countertop blender in batches.
07 - Return bisque to the pot. Stir in apple cider vinegar, heavy cream, and reserved lobster meat. Simmer gently for 5 minutes. Season with salt and pepper. Ladle into bowls, garnish with chives or parsley, and serve hot.