Melt 1 tablespoon of butter in a large heavy-bottomed pot over medium-high heat. Add the reserved shrimp shells and cook for 3 to 4 minutes, stirring frequently, until bright pink and aromatic.

Pour in 1/4 cup brandy or dry sherry and simmer for 1 minute, scraping up any browned bits from the bottom of the pot.

Add the remaining 2 tablespoons of butter, then add diced onion, celery, and carrots. Sauté for 5 to 6 minutes until vegetables soften and turn translucent. Add minced garlic and cook for 1 more minute.

Stir in 2 tablespoons of tomato paste and cook for 1 minute, coating the vegetables and deepening the color.

Pour in 4 cups of seafood or chicken stock. Add 1 bay leaf and 1 teaspoon dried thyme. Bring to a gentle simmer and cook for 15 to 20 minutes.

Strain the mixture through a fine mesh strainer into a bowl, pressing gently on solids to extract liquid. Discard shells and vegetables.

Return the strained broth to the pot. Use an immersion blender to puree for 1 minute until completely smooth.

Over medium heat, add peeled shrimp to the smooth broth. Simmer for 3 to 4 minutes until shrimp turn opaque and pink. Slowly stir in 1 cup heavy cream.

Season with salt, black pepper, and a pinch of cayenne pepper to taste. If bisque is too thin, whisk together 2 tablespoons rice flour or cornstarch with 3 tablespoons cold water, then stir into the soup and simmer for 1 minute to thicken. Keep at a gentle simmer without boiling.