Cook lobsters in boiling salted water for 8-10 minutes until bright red. Remove, cool, extract meat from shells, chop into bite-sized pieces, and refrigerate. Reserve all shells.
Heat olive oil in a Dutch oven, add lobster shells and cook for 3-4 minutes until fragrant. Add water, bay leaf, thyme, and salt. Simmer for 20-25 minutes, then strain through a fine mesh strainer. Set stock aside.
In the same pot, melt 3 tablespoons butter and sauté onion, carrots, and celery for 8-10 minutes until soft. Add garlic, smoked paprika, thyme, tarragon, and cayenne. Cook until fragrant, then stir in tomato paste and cook for 2-3 minutes.
Sprinkle flour over vegetables, stir and cook for 2 minutes. Deglaze with sherry and white wine, cooking until mostly evaporated. Slowly pour in lobster stock with a bay leaf, bring to a boil, then simmer for 15-20 minutes.
Remove bay leaves and thyme sprigs. Blend soup with an immersion blender until smooth. Strain through a fine mesh strainer back into the pot to achieve a silky texture.
Return strained bisque to low heat. Stir in warmed heavy cream and remaining tablespoon of butter. Season with salt, white pepper, and sherry vinegar or lemon juice. Add reserved lobster meat and heat gently for 3-5 minutes without boiling.
Ladle into warmed bowls, arrange extra lobster pieces in the center, swirl with cream if desired, and garnish with fresh chives or parsley. Serve immediately.