1. Heat Broth Mixture: Bring broth, ginger, garlic, and soy sauce to a boil, covered, in a large saucepan; set aside.

2. Season and brown chicken: Meanwhile, pat chicken dry with paper towels, then season with salt and pepper.  Heat oil in large Dutch oven over medium-high heat until just smoking.  Brown chicken lightly on both sides, about 5 minutes total.  Transfer chicken to plate.

3. Brown onion: Add onion and 1/8 tsp salt to fat left in dutch oven and cook over medium-high heat until lightly browned, about 5 minutes.

4. Add broth mixture and cook: Turn heat to low and strain broth into dutch oven, scraping up any browned bits.  Add bok choy stalks and browned chicken.  Cook, covered, until chicken is cooked through and no longer pink in the center, about 10 minutes.

5. Remove chicken and add noodles: Remove chicken from the pot and stir in noodles.  Increase heat to medium-high and cook until noodles are just tender, about 3 minutes.

6. Shred chicken and finish: While noodles are cooking, cut or shred chicken into bite-sized pieces.  Stir in bok choy greens and cook until just wilted, about 1 minute.  Off heat, stir in shredded chicken and green onions.  Season with salt and pepper to taste; serve.