In a small bowl, toss the sliced beef with 1 tablespoon of soy sauce and half of the chili flakes. Set aside.
Heat 1 tablespoon of oil in a large pot or Dutch oven over medium-high heat. Add the beef in a single layer and sear until browned, about 2-3 minutes. Remove beef and set aside.
In the same pot, add the remaining oil. Sauté the mushrooms until browned and tender, about 4-5 minutes.
Add the minced garlic and ginger to the mushrooms, stirring constantly for 1 minute until fragrant. Be careful not to burn the garlic.
Pour in the beef broth and remaining soy sauce. Bring to a boil, then reduce heat to low and simmer for 10 minutes to meld the flavors.
While the broth simmers, cook the noodles in a separate pot of boiling water according to package directions. Drain and rinse.
Add the baby bok choy and the cooked beef (with its juices) back into the broth pot. Simmer for 2-3 minutes until the bok choy is tender but crisp.
Divide the cooked noodles into bowls. Ladle the hot broth, beef, and vegetables over the noodles.
Garnish with chopped green onions and extra chili flakes before serving.