Cook carrots, snow peas, and noodles in boiling water 3 minutes; drain.
Heat 2 teaspoons oil in a large nonstick skillet over medium-high heat. Cut chicken crosswise into thin strips. Add chicken and 1 tablespoon soy sauce to pan; stir-fry 3 minutes. Remove chicken from pan; keep warm.
Combine remaining 2 tablespoons soy sauce, broth, oyster sauce, sugar, and pepper, stirring well. Heat remaining 1 teaspoon oil over medium-high heat. Add mushrooms and ginger to pan; stir-fry for 3 minutes. Add broth mixture, and cook for 1 minute. Add noodle mixture and chicken to pan; cook 1 minute, tossing to combine. Sprinkle with onions and peanuts.