Heat oil in a large pot over medium-high heat and brown chicken for 3 minutes per side. Add ginger and garlic for 30 seconds.
Add broth, soy sauce, mirin and mushrooms. Simmer gently for 15 to 18 minutes until the chicken reaches 165°F, then lift the meat out to slice.
Meanwhile, place eggs in a small saucepan and cover with water by 1 inch. Bring to a boil, reduce to a steady simmer and cook for 10 minutes; cool in cold water, peel and halve.
Cook ramen noodles in a separate pot according to the package and drain. Add bok choy to the broth and simmer for 2 to 3 minutes.
Stir sesame oil into the broth off the heat. Divide noodles and sliced chicken among four bowls, ladle broth and vegetables over them and add two egg halves to each.