01 - Steam or boil live lobsters for 8 to 10 minutes until bright red. Cool, then extract meat from tails, claws, and knuckles. Reserve shells; chop meat into bite-size pieces and refrigerate.
02 - Heat butter and olive oil in a large pot over medium heat. Add lobster shells, onion, carrot, celery, and garlic. Sauté 8 to 10 minutes, stirring occasionally, until vegetables soften and shells release fragrance.
03 - Stir in tomato paste and cook for 2 minutes. Add white wine and cognac, scraping browned bits from pot bottom. Simmer for 2 minutes to reduce alcohol.
04 - Add fish stock, bay leaf, thyme, and paprika. Bring to a boil, then reduce heat and simmer uncovered for 25 minutes to develop flavor.
05 - Remove lobster shells and bay leaf, discarding them. Blend soup until smooth using an immersion blender or regular blender in batches.
06 - Return soup to pot, stir in heavy cream, and gently bring to a simmer. Add lobster meat and heat for 2 to 3 minutes. Adjust seasoning with salt and pepper.
07 - Ladle bisque into bowls. Optionally garnish with chopped chives or parsley. Serve immediately while hot.