I make this every time we get live lobsters, which is only a few times a year. Over the years I have tweaked this. I always have to buy an extra lobster or two tails to add some meat to the soup. It is the best soup I have ever tasted. While it is rich, a single bowl and some bread make a fine supper with champagne.

Lobster Bisque 
Serves 4

Ingredients:

• 3 1- 2 pound live lobsters

For the stock:

• Lobster shells
• 1 onion quartered
• 2 carrots
• 2 cups of lobster cooking water
• Water to cover shells

For the Bisque

• 3 tablespoons olive oil
• 1 onion, large dice (I use a sweet onion like Vidalia)
• 1 large celery stalk, rough chopped
• 1 large carrot, sliced
• 12 garlic cloves, peeled
• 20 grape tomatoes (or one large tomato rough chopped)
• 2 tablespoons dried tarragon
• 2 tablespoons dried thyme
• 2 bay leaves
• 8-10 whole peppercorns
• 1/2 cup brandy
• 1/2 cup dry Sherry (plus more for serving)
• 4 cups lobster stock
• 1 small can tomato paste
• 1 cup whipping cream
• 2 teaspoons cornstarch
• 1 tablespoon water
• Lobster Stock (about 4 cups)
• Chives and fresh black pepper for serving

Method:

To make stock:

• Bring large pot of water to boil.
• Add 1/4 sea salt.
• Add lobsters head first and boil until cooked through, about 15 minutes. Using tongs, transfer lobsters to large bowl.
• Reserve 2 cups cooking liquid. Cool lobsters

• Working over large bowl to catch juices, cut off lobster tails and claws. Crack tail and claw shells and remove lobster meat. Coarsely chop lobster meat; cover and chill. Coarsely chop lobster shells and bodies; transfer to medium bowl. Reserve juices from lobster in large bowl.Or you can serve the lobsters and save the shells, reserving extra meat for the bisque.
• In a clean pot add shells, onion, carrot and reserved cooking liquid. Add enough water to cover the shells with an additional 2″ of water.
• Cook low and slow for several hours (I have a simmer burner and let it simmer over night), until the stock has reduced to the level of the shells. Strain over a large pot.
• You can cool and save this in the freezer or refrigerator or use straight away.
• Discard solids

Bisque:

• Heat olive oil in heavy large pot over high heat.
• Add onion and next 8 ingredients. Boil until almost all liquid has evaporated, about 4 minutes. *Note, if you are using tails for the meat, put them in with veggies and cook till meat is white, about 4 minutes Use one tail per serving. 
• Add lobster stock
• Simmer 1 hour.
• Remove bay leaves
• Use a stick blender to puree
• Strain soup through sieve set over a pot, pressing firmly on solids. Whisk tomato paste into soup. Add sherry and cognac.
• Simmer until soup is reduced to 3 cups, about 15 minutes. (Can be prepared 1 day ahead. Cover and refrigerate at this point)
• Add cream to soup and simmer 5 minutes.
• Dissolve cornstarch in 1 tablespoon water.
• Add to soup and boil until slightly thickened, about 2 minutes. 
• Ladle soup into bowls and top with a pile of lobster meat, some fresh chives, freshly cracked black pepper and a drizzle of sherry.