Every time we have lobster, I make lobster bisque from the carcasses and shells. According to my wife, lobster bisque is actually better than the lobster itself (which is mighty fine in its own right) because of the concentrated lobster taste. It is also much easier to eat.
I do not follow a particular recipe but I am sure all lobster bisque recipes must be similar. I use only a small amount of flour to thicken the soup as I do not want a very thick starchy bisque. Some may like to use all heavy cream to make the bisque thicken but, for me, that is a bit too much.
Lobster Carcasses: from 2 lobsters, cut into manageable size.
Vegetables: onion (2, finely diced), celery (3 stalks finely sliced), carrots (3, cut into small chunks).
Tomato paste: about 2 tbs
White wine: 1 cup
Water to just cover the vegetables and lobster carcasses.
Thyme: 1/4 tsp, dried
Bay leaves: 2-3
Olive oil and butter.
Flour: 2-3 tabs (to thicken the bisque)
In a deep stock pot, I added butter and olive oil (1 tbs each). When hot, I added the onion and celery and sautéed until soft (3-4 minutes) and seasoned it with salt and pepper. I moved the vegetables aside and added the tomato paste to the bottom of the pan and moved it around to caramelize it as indicated when the color turned dark. This took about 1-2 minutes. Then I added the white wine (left over from dinner the previous night) and simmered for few minutes. I then added the carrots and the lobster carcasses. Using a wooden spatula, I pushed down on the carcasses and then poured in the water (we always use filtered water from our reverse osmosis system) to just cover. I added the thyme and bay leaves. I put the lid on and let it simmer for one hour or more. I occasionally mixed/pressed down on the carcasses using a wooden spatula during the cooking to make sure everything was submerged.
Using a large fine meshed sieve, I strained and removed all solids. This yielded about 4 cups of broth. I let it cool down to the room temperature, covered, and placed in the refrigerator (of course, you can do the reduction part without a pause).
Next day, I melted butter (about 3 tbs) in a stock pot. When melted, I added the flour (3 tbs) and mixed until no raw flour was visible but not colored (about 5 minutes). I added about a cup of cold broth and whisked quickly. When the flour/butter mixture was completely incorporated, I added the rest of the broth to the pot. At this point, the broth was very slightly thickened. I let it simmer without the lid on for about one hour to reduce it to half (about 2 cups). At the end of the reduction, I tasted it. Because of the saltiness from the lobster carcasses and the reduction, no salt was needed. The broth was nicely seasoned and all the essence of lobster was there. Again, I let it cool down to room temperature and place the reduced broth into a sealable container and placed it in the refrigerator.
To serve, I poured the amount for two servings into a sauce pan and gently heated it up. When warm, I added cream (about 4 tbs, the amount is up to you) and gently stirred. I do not like very thick starchy bisque but I also do not like too much heavy cream in my bisque. I poured it into soup bowls with the cooked lobster meat in the center. I did not add any other garnish. I even omitted the slug of sherry usually included in this kid of bisque by my wife's request. She said she did not want to risk having the sherry overpower the lobster flavor.
As my wife said, this is a very essence of the lobster. It is velvety smooth with all the flavor of lobster. The lobster meat got warmed up which was very sweet.