1. Make stock: Heat star anise, cinnamon, and cloves in a small dry skillet over medium-high heat until fragrant, about 2 minutes. Transfer to a 6-quart stockpot.

2. Preheat broiler. Broil ginger and onions, flipping once, until blackened, about 5 minutes per side. Transfer to pot. Add oxtail, 2-inch scallion pieces, and peppercorns. Add 5 quarts water; bring to a boil. Skim foam. Add salt. Reduce heat. Simmer, skimming occasionally, 2 1/2 hours.

3. Pour stock through a large sieve into a large bowl; discard solids. Let cool 20 minutes. Pour through a cheesecloth-lined sieve into a large bowl. Refrigerate, covered, 6 hours or overnight.

4. Make soup: Chill beef in freezer until firm, about 2 hours. Cover noodles with cold water. Let stand until noodles are softened, about 30 minutes; drain.

5. Cut beef in half. Place each half flat side down, and cut beef against the grain as thinly as possible. Allow beef to warm to room temperature.

6. Skim fat from stock; discard. Transfer stock to a pot; add shallots, and bring to a boil. Reduce heat, and simmer until shallots are soft, about 15 minutes.

7. Bring a medium pot of water to a boil. Add noodles; cook until just tender, about 10 seconds; drain.

8. Divide noodles, beef, and sprouts among 4 bowls. Add simmering stock (it will cook beef gently). Top with thinly sliced scallions and herbs; serve each with a lime wedge.