Combine the first 5 ingredients in a large zip-top plastic bag, and seal. Marinate in refrigerator 2 1/2 hours, turning bag occasionally.
Heat a small Dutch oven over high heat. Coat pan with cooking spray. Add beef mixture to pan; stir-fry 1 minute or until browned. Remove beef mixture from pan; set aside.
Add bok choy, mushrooms, carrot, and green onions to pan; stir-fry 2 minutes or until bok choy begins to wilt. Add water, hoisin, and broth; bring to a boil. Stir in noodles. Reduce heat; simmer 5 minutes or until noodles are done. Stir in beef mixture and vinegar. Ladle 1 1/2 cups soup into each of 6 bowls; drizzle each serving with 1/4 teaspoon sesame oil.