01 - Bring a large pot of salted water to a rolling boil. Add live lobsters and cook for 7-8 minutes until shells turn bright red. Transfer to an ice bath to cool immediately. Remove meat from claws, tails, and knuckles, reserving shells. Chop meat into bite-sized pieces and refrigerate until needed.
02 - Heat butter and olive oil in a large pot over medium heat. Add onion, carrot, celery, and garlic. Sauté for 6-8 minutes, stirring occasionally, until vegetables are softened and fragrant but not browned.
03 - Stir in tomato paste and cook for 2 minutes to develop depth. Add reserved lobster shells and continue stirring for 2 minutes to coat and toast slightly, extracting maximum flavor.
04 - Pour in white wine and scrape the bottom of the pot with a wooden spoon to release any browned bits. Simmer for 2 minutes to allow alcohol to evaporate and concentrate flavors.
05 - Add seafood stock, bay leaf, thyme, and cayenne pepper. Bring to a boil, then reduce heat to low and simmer uncovered for 30 minutes, allowing flavors to meld and broth to reduce slightly.
06 - Remove and discard lobster shells and bay leaf. Use an immersion blender to purée the soup until completely smooth. Alternatively, transfer to a countertop blender in batches, being cautious with hot liquid.
07 - Return bisque to the pot over low heat. Stir in apple cider vinegar, heavy cream, and reserved lobster meat. Simmer gently for 5 minutes to heat through. Taste and adjust seasoning with salt and freshly ground black pepper as needed.
08 - Ladle hot bisque into warmed bowls. Garnish with finely chopped fresh chives or parsley. Serve immediately with crusty bread for a complete experience.