Heat oil in large pot over medium-high; brown chicken 3 minutes per side. Add ginger and garlic 30 seconds.
Add broth, soy sauce, mirin, and mushrooms; gently simmer 15 to 18 minutes until chicken 165°F. Remove chicken to slice.
Meanwhile, cover eggs with water by 1 inch in small saucepan. Boil; reduce to steady simmer and cook 10 minutes. Cool in cold water, peel, and halve.
Cook noodles separately per package; drain. Add bok choy to broth; simmer 2 to 3 minutes.
Off heat, add sesame oil. Divide noodles and sliced chicken among four bowls; ladle broth and vegetables over, adding two egg halves each.