To prepare beef, sprinkle cornstarch over beef; toss to combine. Add 1 tablespoon soy sauce and next 4 ingredients (1 tablespoon soy sauce through 3 garlic cloves); toss well to coat. Cover and refrigerate 30 minutes to 1 hour.
Heat a large nonstick skillet or wok coated with cooking spray over medium-high heat. Add beef mixture; stir-fry 3 minutes or until done. Remove mixture from pan. Cover and keep warm.
To prepare noodles, pour boiling water over noodles; let stand 10 minutes or until tender. Drain and rinse with cold water. Drain. Snip noodles several times with kitchen shears.
To prepare vegetables, wipe skillet or wok clean with paper towels. Heat 1 teaspoon sesame oil and vegetable oil in pan over medium-high heat. Add red pepper and 5 garlic cloves; stir-fry 30 seconds. Add mushrooms, onions, and carrot; stir-fry 3 minutes. Add half of spinach; stir-fry 2 minutes or until spinach wilts. Add remaining spinach; stir-fry 2 minutes or until spinach wilts.
Reduce heat to medium-low. Add beef mixture and noodles to pan, stirring well to combine. Combine 1/3 cup soy sauce, brown sugar, vinegar, and 1 tablespoon sesame oil, stirring with a whisk. Drizzle over noodle mixture; stir well to combine. Cook over medium-low heat 3 minutes or until thoroughly heated. Sprinkle with sesame seeds.