Peel and devein shrimp, reserving shells. Cover and refrigerate shrimp. (Make-ahead: Refrigerate for up to 8 hours.)
In large shallow saucepan, heat butter over medium heat; cook shallots, stirring, until softened, about 3 minutes. Add tomato paste and paprika; cook, stirring, for 1 minute. Add reserved shrimp shells; cook, stirring, until pink, about 2 minutes.
Add wine and 2 cups water, stirring to scrape up brown bits. Bring to boil; reduce heat and simmer gently for 20 minutes. Into clean saucepan, strain through fine-mesh sieve, pressing shells to extract liquid. Discard shells. (Make-ahead: Let cool for 30 minutes; refrigerate until cold. Cover and refrigerate in airtight container for up to 8 hours. Reheat to continue.)
Add cream and pepper; bring to boil. Boil, stirring, until reduced by half, about 10 minutes. Add reserved shrimp; cook until pink, about 5 minutes. Stir in parsley.
RICE: Meanwhile, in saucepan, heat butter over medium heat; cook shallot and carrot until softened, about 5 minutes. Stir in rice. Add 4 cups water and salt; bring to boil. Reduce heat to medium-low; cover and cook until water is absorbed and rice is tender, about 20 minutes. Fluff with fork. Serve shrimp mixture over rice. Garnish with chives.