Heat 1 tablespoon oil in a large saucepan or Dutch oven over medium-high heat. Add scallion whites, garlic and ginger and cook, stirring, until fragrant, about 1 minute.
Add water, miso, soy sauce and the remaining 1 tablespoon oil; bring to a boil.
Stir in chicken, bok choy, noodles and mushrooms. Cover and return to a boil.
Uncover, reduce heat to medium and cook, stirring, until the chicken is cooked through and the vegetables are tender, 3 to 5 minutes.
Serve sprinkled with the scallion greens.
