01 - Bring a large pot of salted water to a boil with the bay leaf and peppercorns. Add live lobsters, cover, and cook for 6–8 minutes until bright red. Remove lobsters with tongs and set aside to cool. Reserve ½ cup of the cooking liquid. Twist off tails and claws, crack shells, and remove lobster meat; chop into bite-sized pieces. Set aside. Save shells for the bisque base.
02 - In a large pot, heat olive oil over medium heat. Add onion, celery, carrots, leek, and fennel. Cook for 5–7 minutes until softened. Stir in garlic and tomato paste; cook for 2 minutes. Add lobster shells, thyme, and tarragon. Sauté for 5 minutes, stirring occasionally.
03 - Pour in brandy (or cognac) and ignite carefully to flambé, letting the alcohol burn off. Once the flames subside, add white wine and reserved lobster cooking liquid. Pour in fish stock. Bring to a boil, then reduce heat and simmer uncovered for 30–40 minutes.
04 - Remove from heat. Discard herb sprigs. Use an immersion blender to blend the mixture, then strain through a fine sieve into a clean pot, pressing firmly to extract all liquid. Discard solids.
05 - Bring strained bisque to a gentle simmer. Stir in heavy cream and season with salt, pepper, and cayenne. Add chopped lobster meat, reserving a few pieces for garnish. Simmer for 2–3 minutes to heat through. Taste and adjust seasoning with lemon juice as needed.
06 - Ladle bisque into warm bowls. Top with reserved lobster pieces and a few tarragon leaves.