Bring 3 cups of water to a boil in a large pot. Add lobster tails and cook for 5-6 minutes until the shells turn bright red.

Remove the lobster from the pot and save the cooking water for the stock.

Let the lobster cool, then remove the meat from the shells. Chop it into small bite-sized pieces and set aside. Save the shells!

Heat 1 tbsp olive oil in a large pot over medium heat. Add the lobster shells and sauté for 2-3 minutes to release their flavor.

Stir in onion, garlic, celery, and carrot, cooking until soft (about 5 minutes).

Add tomato paste, salt, black pepper, paprika, and cayenne (if using). Cook for another 1-2 minutes.

Pour in ½ cup white wine, scraping the bottom of the pot to deglaze. Let it simmer for 2 minutes.

Add the reserved lobster cooking water and bring to a boil. Reduce heat and simmer uncovered for 20 minutes.

Strain the stock through a fine-mesh sieve into a bowl, discarding the solids.

In the same pot, melt 2 tbsp butter over medium heat. Add 2 tbsp flour and whisk constantly for 1-2 minutes until it turns golden.

Slowly pour in the strained lobster stock, whisking to avoid lumps. Simmer for 5 minutes until slightly thickened.

Stir in 1 cup heavy cream and ½ cup whole milk. Simmer for another 5 minutes.

Season with ½ tsp salt, ½ tsp white pepper, and ¼ tsp nutmeg.

Use an immersion blender to puree the soup until silky smooth. Alternatively, blend in batches in a stand blender.

Return the bisque to the pot and stir in 1 tbsp cognac or brandy (optional but recommended).

Stir in the chopped lobster meat and let it warm through for 2-3 minutes.

Taste and adjust seasoning if needed. If too thick, add a little extra broth or milk to thin it out.

Finish with a squeeze of fresh lemon juice for brightness.

Ladle the bisque into bowls.

Garnish with chopped chives or parsley.

Serve with crusty bread or oyster crackers for dipping.