Melt 1/2 cup butter in a heavy-bottomed stock pot or dutch oven over medium-high heat. Cook and stir the leek, carrot, and celery in the melted butter until tender, 5 to 7 minutes.
As the vegetables cook, melt 1/2 cup butter in a skillet over medium heat. Make a roux by cooking and stirring the flour into the butter until just a little golden, just enough to cook off the flour taste. This is not a heavy gumbo roux.
Stir the tarragon, thyme, bay leaf, tomato paste, and cayenne pepper to the vegetable mixture; stir to distribute evenly. Pour the brandy over the mixture; cook and stir until the heat cooks off most of the brandy, 2 to 3 minutes. Pour 1 cup of the fish stock over the mixture.
Pour the hot mixture into a blender, filling the pitcher no more than halfway. Hold the lid of the blender in place with a folded kitchen towel and carefully start the blender, using a few quick pulses to get the mixture moving before leaving it on to puree. Puree in batches until smooth and pour back into the stock pot. Alternately, you can use a stick blender and puree the mixture in the pot without using the blender.
Stir in the roux and 1 cup more of the fish stock to the pureed mixture; return to heat, stirring well.
Blend the diced tomatoes and about 3/4 of the crawfish tails together in a blender until smooth; stir into the stock pot. Add the remaining fish stock; stir. Bring the mixture to a boil, reduce heat to medium-low, and simmer another 2 to 3 minutes. Stir the heavy cream through the liquid; return to a simmer, but do not allow to come to a boil. Season with the salt and white pepper. Serve hot in bowls garnished with the remaining crawfish tails and the paprika.
