Cook noodles according to package directions; drain.
Place garlic, pepper, and broth in a large saucepan. Bring to a boil; reduce heat, and simmer 10 minutes.
Combine soy sauce, sake, and honey in a small bowl; stir with a whisk.
Heat a large nonstick skillet coated with cooking spray over medium-high heat. Add mushrooms and carrot; sauté 2 minutes. Stir in soy sauce mixture; cook 2 minutes, stirring constantly. Add vegetable mixture to broth mixture. Stir in beef; cook 2 minutes or until beef loses its pink color. Stir in noodles, green onions, and spinach. Serve immediately.