01 - Bring a large pot of salted water to a boil. Add live lobsters and cook for 8–10 minutes until bright red. Remove and let cool. Extract meat from tails, claws, and knuckles. Reserve shells. Chop meat into bite-sized pieces; set aside.
02 - In a large pot, heat butter and olive oil over medium heat. Add onion, carrot, and celery. Sauté for 6–8 minutes until softened but not browned.
03 - Stir in garlic and tomato paste; cook for 1 minute. Add reserved lobster shells, bay leaf, thyme, and paprika. Cook for 2–3 minutes, stirring frequently.
04 - Pour in white wine and apple cider vinegar. Simmer for 2 minutes, scraping up any browned bits from the bottom.
05 - Add seafood stock. Bring to a boil, then reduce heat and simmer uncovered for 25 minutes to extract maximum flavor from shells.
06 - Remove and discard lobster shells and bay leaf. Using an immersion blender, purée the soup base until very smooth. Alternatively, blend carefully in batches using a countertop blender.
07 - Stir in heavy cream and milk. Add chopped lobster meat, reserving a few pieces for garnish if desired. Simmer gently for 5 minutes. Season with salt, pepper, and cayenne to taste.
08 - Ladle bisque into warm bowls. Garnish with reserved lobster pieces and chopped chives or parsley.