Peel and devein the shrimp, reserving the shells. Pat the shrimp dry, cover, and refrigerate.

Melt 2 tbsp butter in a large pot over medium heat. Cook the onion, celery, carrot, and a pinch of salt until softened, about 7 minutes. Add the garlic, tomato paste, and paprika; stir for 1 minute.

Sprinkle in the flour and stir for 1 minute. Pour in the sherry and cook, scraping the pot, for 30 seconds. Add the shrimp shells, 4 cups water, and a large pinch of salt. Simmer gently for 20 minutes.

Remove and discard the shells. Blend the soup in batches until smooth, then return it to the pot.

Stir in the cream and cayenne. Bring to a bare simmer, add the shrimp, and poach until opaque, 3 to 4 minutes.

Remove from heat, stir in the remaining 2 tbsp butter, then ladle into bowls and serve with black pepper.