Melt a 1/4 cup of the butter in a large soup pot over medium heat. Add the onion, carrot, celery, and tomato and cook, covered, stirring occasionally, until soft, about 12 minutes. Raise the heat to high. Add the shrimp shells and rice, and cook, stirring, until deep red in color, about 2 minutes. Add the tomato paste and cook, stirring, until caramelized, about 2 minutes more. Remove the pot from the heat, pour in the 1/2 cup of the Cognac, return to a high heat and cook until almost dry, about 1 1/2 minutes.

Tie the parsley sprigs, thyme, and bay leaf together with a piece of kitchen sting and add to the pot. Pour in the water and 1 1/2 tablespoons salt, and bring to a boil. Lower the heat and simmer, covered, for 30 minutes. Remove from the heat and allow to cool.

Remove and discard the herb bundle. Working in batches, transfer the mixture to a blender and puree until smooth. Strain the puree through a fine sieve into a pot and discard the solids.

Heat the bisque over medium heat. Whisk in the remaining 1/4 cup butter, remaining 2 tablespoons Cognac, heavy cream, lemon juice, and cayenne into the soup and season with salt and pepper to taste. Divide among soup bowls, garnish with croutons, and serve immediately.

Difficulty: Easy