Cook noodles according to package directions; drain.
Combine sake and the next 4 ingredients (sake through crushed red pepper) in a small bowl; set aside.
Heat 2 teaspoons of vegetable oil in a large nonstick skillet over medium-high heat. Add chicken; stir-fry 4 minutes or until done. Remove chicken from pan; keep warm. Heat 1 teaspoon vegetable oil and sesame oil in pan over medium-high heat. Add mushrooms and remaining ingredients; stir-fry 4 minutes or until broccoli is crisp-tender. Return chicken to pan; stir in sake mixture. Cook 1 minute or until thoroughly heated. Serve over noodles.