• Fill a 6-quart kettle three fourths full with salted water and bring to a boil. Plunge lobster headfirst into water and cook, covered, over high heat 8 minutes.
• Transfer lobster with tongs to a large bowl. Let lobster stand until cool enough to handle.
• Reserve 2 cups cooking liquid. 

2. Process lobster

Over a bowl to catch the juices, twist off tail and claws and reserve juices.

Remove meat from the tail, reserving shells and lobster body. (Lobster meat will not be cooked through.)  Discard internal organs (traditionally eaten but now known not to be good for you).

Coarsely chop meat and transfer to a bowl. Chill lobster meat, covered.

3. Prep Soup

• Chop onion, celery, carrot, and tomato and halve garlic head crosswise. In a 6-quart heavy kettle heat oil over moderately high heat until hot but not smoking and sauté reserved lobster shells and body, occasionally stirring, 8 minutes.
• Add vegetables, garlic, herbs, peppercorns, brandy, and Sherry and simmer, stirring, until almost all liquid is evaporated, about 5 minutes.
• Add lobster juices and cooking liquid. Simmer mixture, uncovered, stirring occasionally, 1 hour.

4. Finish Soup

• Pour mixture through a fine sieve into a large saucepan, pressing on and discarding solids.
• Whisk in tomato paste and simmer until reduced to about 3 cups, about 10 minutes.
• Add cream and simmer bisque 5 minutes.
• In a small bowl, stir together cornstarch and water and whisk into bisque. Simmer bisque, stirring, 2 minutes. (Bisque will thicken slightly.)
• Add lobster meat with any reserved juices and simmer bisque 1 minute, or until lobster meat is just cooked through — season bisque with salt and pepper.

Adapted from: Epicurious.com