In a medium pot, combine stock with two pieces of scallion, beef bones, ginger, and garlic. Simmer covered for 40 minutes. Strain the liquid and transfer back in the same pot. Reduce the liquid in half. Season with salt and pepper.
Skim the fat from the surface of the broth.
In a pot of salted boiling water, cook noodles following package instructions. Quickly blanch into an ice water bath, then squeeze out excess water.
Arrange noodles in shallow bowls, garnish with cabbage, eggs, beef slices, and remaining slices scallions.
Serve the hot broth on the side.