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 - Heat butter and olive oil in a large pot over medium heat. Add onion, carrot, celery, and garlic. Sauté for 6-8 minutes until vegetables are softened and fragrant.
03 - Stir in tomato paste and cook for 2 minutes to caramelize. Add reserved lobster shells and continue stirring for another 2 minutes to extract flavors.
04 - Pour in white wine, scraping browned bits from the bottom of the pot. Simmer for 2 minutes to reduce slightly and concentrate flavors.
05 - Add seafood stock, bay leaf, thyme, and cayenne pepper. Bring to a boil, then reduce heat and simmer uncovered for 30 minutes to develop depth.
06 - Remove and discard lobster shells and bay leaf. Use an immersion blender to purée until completely 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 to taste.
08 - Ladle hot bisque into bowls. Garnish with finely chopped fresh chives or parsley. Serve immediately.