Peel the shrimp, placing the shells in a medium saucepan. Set the peeled shrimp aside on a cutting board. Add the broth to the saucepan with the shells.
Bring the broth and shells to a simmer over medium heat. Reduce heat to low, cover, and let simmer gently for 30 minutes. Strain the broth through a fine-mesh sieve into a bowl, pressing on the shells to extract all liquid. Discard the shells.
While the stock simmers, chop the peeled shrimp into bite-sized pieces and refrigerate until needed.
In a large pot or Dutch oven, melt the butter over medium heat. Add the shallots and cook until softened, about 5 minutes. Stir in the garlic and cook for 1 more minute until fragrant.
Sprinkle the flour over the shallots and cook, stirring constantly, for 2 minutes to form a roux. Add the tomato paste and continue to stir for another 2 minutes.
Whisk in the dry sherry, scraping up any browned bits from the bottom of the pot. Cook until the sherry has mostly evaporated.
Gradually pour in the reserved shrimp stock while whisking constantly to prevent lumps. Whisk in the half-and-half. Bring the soup to a gentle simmer and cook, stirring occasionally, until it has thickened, about 5-7 minutes.
Stir in the chopped shrimp. Cook just until the shrimp are pink, opaque, and cooked through, about 3-4 minutes. The internal temperature of the shrimp should reach 145°F (63°C).
Season with salt and pepper to taste. Serve hot, garnished with fresh chives.