Peel and devein half of the shrimp (about 8–12), including the tails. Cut them into bite-sized pieces and refrigerate until needed. Reserve the shells.
Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the shell-on shrimp and reserved shells, stirring constantly until they turn bright pink, about 2 minutes. Remove from the pot and set aside.
Melt the butter in the same pot, then add the onion, celery, carrots, and garlic. Lower the heat to medium. Season with salt and white pepper, and cook, stirring frequently, until the vegetables are softened but not browned, about 8 minutes.
Stir in the tomato paste, paprika, and cayenne. Cook for 1–2 minutes, until the tomato paste darkens slightly. Add the flour, stirring until the vegetables are evenly coated. Turn off the heat.
Return the cooked shrimp and shells to the pot. Add the white wine and brandy, turn the heat back up to medium-high, and stir until most of the liquid has evaporated.
Add the water, bay leaf, and thyme sprig, season again with salt and white pepper, then cover and simmer for 20 minutes.
Remove the thyme and bay leaf. Puree the soup until very smooth using an immersion blender or a high-powered traditional blender (work in batches and be careful with hot liquids — avoid blenders that don’t release steam, like bullet-style models).
Since the mixture includes shrimp shells, you’ll need to blend thoroughly for several minutes to achieve a smooth texture.
Set a fine-mesh strainer, chinois, or china cap over another large pot or Dutch oven. Pour the soup through and press with a ladle or spatula to extract as much liquid as possible.
Return the strained soup to the stove over medium-low heat. Add the reserved peeled and diced shrimp, then stir in the heavy cream, sherry, and lemon juice. Simmer for 2–3 minutes, just until the shrimp are cooked through.
Taste and adjust seasoning as needed.
Ladle into bowls and garnish, if desired, with a drizzle of cream or a dollop of crème fraîche, and a sprinkle of chives. Serve immediately.