Pat lobster tails dry. Using kitchen shears, cut through the top shell to remove the meat. Chop the meat into bite-size pieces and refrigerate. Save all shells.
Melt butter in a large Dutch oven over medium heat. Add onion, carrots, and celery and cook until softened, about 5 minutes.
Add the reserved lobster shells and tomato paste. Stir and cook 3 minutes to deepen the color and build flavor.
Sprinkle in the flour and stir constantly for 2 minutes to cook out the raw flour taste.
Pour in the cognac (if using) and cook 1 minute, scraping up any browned bits. Add the white wine and sherry and simmer 2 minutes.
Stir in the lobster stock, heavy cream, bay leaves, tarragon, and Italian seasoning. Bring to a gentle boil, then reduce heat and simmer uncovered for 30 minutes.
Remove and discard bay leaves. Use tongs or a slotted spoon to remove as many shells as possible.
Blend the soup until smooth with an immersion blender. (If using a countertop blender, blend in small batches and vent the lid carefully.)
Strain the bisque through a fine-mesh strainer (line with cheesecloth for extra-smooth texture). Return the strained soup to the pot.
Add the reserved lobster meat and warm gently over low heat until just cooked through, 2–4 minutes. Do not boil.
Season with salt, pepper, and optional cayenne. Taste and adjust. Serve hot, garnished with chives and a small swirl of cream if desired.