How to Make It
Peel and devein shrimp, leaving tails on if desired. Refrigerate shrimp until ready to use, and reserve shells.
Heat oil in a Dutch oven over high. Add shells; cook, stirring constantly, until bright red and slightly toasty, 3 to 4 minutes. Reduce heat to medium. Add tomato paste, onion, carrot, and celery; cook, stirring constantly, until fragrant, 2 to 3 minutes. Add 6 cups water and bay leaf; increase heat to high. Bring to a boil; cover and reduce heat to medium-low. Simmer until fragrant and rich, about 50 minutes. Pour through a fine wire-mesh strainer into a bowl, pressing slightly to extract liquid. (You should have 4 cups of liquid.) Discard solids.
Add shrimp liquid, chicken stock, rice, miso, and sherry to a stockpot. Bring to a boil over high. Cover, reduce heat to medium-low, and simmer until rice is very tender, 25 to 30 minutes. Working in batches if needed, transfer mixture to a blender. Remove center piece of blender lid (to allow steam to escape); secure lid on blender, and place a clean towel over opening in lid. Process until smooth, about 60 seconds. Stir in vinegar and cayenne. Cover and keep warm.
Heat butter in a large skillet over high until foamy. Add shrimp; cook, stirring often, until just translucent, about 3 minutes. Add 1 tablespoon dill; cook, stirring often, 1 minute. Serve soup in shallow bowls; top evenly with shrimp, remaining 1 tablespoon dill, and black pepper, if desired.