Add 1 tablespoon of butter to a large pot over medium heat and cook the shrimp shells until they are pink in color. See note.
Set the shells to the side in a bowl and the remaining ½ tablespoon of butter to the pot over medium heat and in the onions, celery, fennel, carrots, bay leaf, parsley, thyme, and tarragon, and cook until lightly browned about 4-5 minutes. Stir occasionally.
Mix in the tomato paste and cook over low heat until fragrant, about 2-4 minutes.
Deglaze by adding in the wine and cognac and cook until the liquid has been mostly absorbed into the vegetables and evaporated, which takes about 3-4 minutes. There should be 1 - 2 tablespoons remaining.
Add the shrimp shells back in along with the water and simmer over low to medium heat for 45-60 minutes.
Roughly blend the shrimp stock with a hand blender or do it in batch in a regular blender.
Strain the soup completely with a fine mesh strainer or chinois into a container and set it aside briefly.
Next, in a separate large pot over low heat make the roux and then pour in the shrimp stock.
Turn the heat to medium-high while continually whisking until it becomes very thick. Turn the heat down to medium and cook for a further 10 minutes.
Finish by whisking in the cream, salt, and white pepper.
Strain through a fine-mesh strainer or chinois until completely smooth.
Serve with optional cooked shrimp, crème Fraiche, or sliced chives
Make-Ahead: This soup is meant to be eaten right away, but you can make it up to 1 day ahead of time. Be sure the cream has not broken from the soup before serving.
How to Reheat: Add the desired amount to a small size pot and cook over low heat while whisking until warm. You can also heat in a microwave until warm.
How to Store: Cover and keep in the refrigerator for up to 4 days. Cover and freeze for up to 3 months. Thaw in the refrigerator for 1 day before reheating and serving.
If you are using pink shrimp shells then they will turn white when cooking them in the first step.
If you don’t want to puree the shells because it weirds you out, back off the 12 cups of water and only put in 8 cups of water to further concentrate the flavor. You will have less soup of course.
You would use this exact same procedure with leftover lobster tails and lobster bodies if wanting to make lobster bisque.
You may need to strain both the stock and the creamed bisque a few times to ensure it is really smooth.
There are some optional garnishes for this soup that include some cooked shrimp, crème Fraiche, thinned sour cream with milk, and thinly sliced chives.
If the cream breaks and the cream separates from the soup, whisk in a small amount of roux over medium heat in a pot to help bring it back together. The same rules with heat and whisking apply when normally working with a roux.
white wine options are chardonnay, pinot grigio, or sauvignon blanc.