Heat the sesame oil and add the ginger, garlic and green onions; cook, stirring often, until fragrant and slightly softened, 2 minutes. Stir in the broth, miso, soy sauce and mirin, stirring until the miso dissolves. Add the mushrooms, sweet potato, turnip and parsnip; top with the chicken. Cover the pan and transfer to the oven. Bake 45 minutes, uncover and continue cooking 15 minutes longer until the chicken and vegetables are very tender. Remove from the oven and let stand 10 minutes before carving chicken. Serve in shallow bowls with the vegetables and broth.