Step 1

Melt 1 tablespoon butter in a large heavy pot over medium-high heat. Add shrimp shells and cook, stirring frequently, until they begin to brown, about 5 minutes. Add 1 bay leaf and 8 cups water. Simmer uncovered for 25 minutes. Set a fine-mesh strainer over a large bowl. Strain, discarding solids. DO AHEAD: Shrimp stock can be made 3 days ahead. Let cool slightly; chill uncovered until cold, then cover and keep chilled.

Step 2

Melt 1 tablespoon butter in a large heavy pot over medium heat. Add shrimp and cook, stirring occasionally, until just opaque in center, 3-4 minutes. Transfer shrimp to a plate. Add remaining 2 tablespoons butter to same pot. Add carrots, celery, and onion and cook over medium heat, stirring frequently, until very soft, about 20 minutes.

Step 3

Remove pot from heat; add brandy. Return pot to heat and stir until almost evaporated, 2-3 minutes. Add rice and tomato paste; stir for 1 minute. Add shrimp stock, remaining bay leaf, parsley, thyme, and 1/4 teaspoon cayenne. Simmer uncovered until flavors meld and rice is soft, about 20 minutes. Season with salt and pepper. Reserve 4 shrimp for garnish and stir remaining shrimp into bisque. Remove parsley, thyme, and bay leaf.

Step 4

Working in batches, purée bisque in a blender until smooth. Pour through a finemesh strainer set over a clean pot. Discard solids in strainer. Stir in cream and reheat bisque over medium heat. Add lemon juice. Season to taste with salt, pepper, and more cayenne, if desired.

Step 5

Mince reserved shrimp and mix with chives in a small bowl. Place 1 heaping tablespoon shrimp mixture in the center of large, shallow soup bowls. Ladle bisque around garnish and serve.