Heat 1 tablespoon of oil in a large saucepan or Dutch oven over medium-high heat. Add scallion whites, garlic and ginger. Cook until fragrant, about 1 minute. Stir while cooking.
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. Cook for 3 to 5 minutes, stirring, until the chicken is cooked through and the vegetables are tender. Serve your miso ramen dish sprinkled with the scallion greens.