1. Bring a large pot of water to boil. Add noodles. Boil until tender (4 minutes). Rinse under cold water; drain. Set aside.
2. Add to pot broth, water, scallion halves, ginger, anise, peppercorns, and 1/2 tsp salt. Boil, then let simmer.
3. Put a large ovenproof skillet in center of oven; heat to 400°F. Pat duck dry. Using tip of knife, score skin in a crosshatch, being careful not to cut into duck; season with 1/4 tsp salt and pepper. Roast, skin down, in skillet until meat thermometer registers 165°F (about 30 minutes).
4. Turn on broiler. Drain fat, turn duck over, and broil 6 inches from heat until skin is golden (2 minutes). Transfer duck to cutting board; let rest 10 minutes. Slice.
5. Strain broth. Return to pot, add sprouts; boil 5 minutes. Arrange noodles, duck, and sliced scallion in bowls. Ladle broth over them. Serve.