Combine first 6 ingredients, stirring with a whisk; set aside.
Heat vegetable oil in a large Dutch oven over medium-high heat; add beef, browning on all sides. Add broth mixture, green onions, garlic, and ginger; bring to a boil.
Cover, reduce heat, and simmer for 1 hour and 30 minutes or until beef is tender. Stir in bok choy and carrot, and cook 5 minutes or until tender. Serve beef mixture over noodles.