Trim the green onions and thinly slice the green parts at an angle. Reserve for serving. Place the whole green onion whites in a 5-to 6-quart Dutch oven along with the chicken, broth, 5 cups water and ginger. Bring to a boil over high heat, cover, and reduce the heat to low. Simmer until the chicken pulls away from the bones, about 15 minutes. Meanwhile, trim and thinly slice the cabbage (you should have about 5 cups).
With a slotted spoon, remove and discard the green onion whites and ginger from the soup. With the same spoon, transfer the chicken to a plate. Cool slightly, then pull the meat off the bones in chunks. Discard the bones. Strain the broth through a fine mesh sieve.
Return the chicken meat and broth to the Dutch oven. Bring to a boil. Add the cabbage, noodles, and snow peas. Cook, stirring occasionally, until the noodles are just tender, about 5 minutes. Stir in the soy sauce, season to taste with salt and pepper, and top with the green onion greens.