Bring large pot of water to boil. Add lobster tails to water, and boil until cooked through and bright red, approximately 5 minutes.
Using tongs, transfer lobsters to large bowl. Reserve 2 cups cooking liquid, saving as much loose lobster meat with the liquid as possible.
Cool lobster tails by running under cool water. Crack tails and remove meat using the method of your choice - see Notes.
Coarsely chop lobster meat, then cover and chill. Coarsely chop lobster shells and transfer to medium bowl. Reserve juices from lobster in a bowl.
In large pot of medium-high heat, heat olive oil. Add lobster shells and sauté until shells darken, approximately 5 minutes.
Add onion, celery, carrots, garlic, tomato, tarragon, thyme, bay leaves, and peppercorns. Stir in brandy and sherry, then boil until almost all liquid has evaporated, approximately 5 minutes.
Add clam juice, reserved liquid from boiling lobster, and reserved lobster juices. Cook at strong simmer for approximately 15 minutes.
Strain soup through sieve set over large saucepan, pressing firmly on solids. Whisk tomato paste into soup. Simmer until soup is reduced to 3 cups, approximately 15 minutes. If making bisque ahead of time, stop here. Cover and refrigerate up to 1 day.
Add whipping cream to soup and simmer 5 minutes. Dissolve cornstarch in 1 tablespoon water. Add to soup. Boil until slightly thickened, approximately 2 minutes.
Season with salt and black pepper. Mix lobster meat into soup and stir until heated through.
Ladle soup into bowls and serve.