Heat sesame oil in a large pot over medium high heat. Add ginger and the white parts of onions and cook for about two minutes until fragrant and starting to brown. Add garlic and cook for 30 seconds more. Add broth and bring to a boil. Cover, reduce heat, and simmer for 5 minutes.
In the meantime, add the eggs to a small saucepan and cover with cold water. Heat over high heat until the water boils, then turn down to a simmer and cook for 6 minutes for soft boiled or 12 minutes for hard boiled. Remove from the heat and run under cold water. Peel the eggs and set aside.
Add soy sauce and brown sugar to the broth mixture and stir to combine. Add sirloin, stir, and cook for about 5 minutes. Add noodles and cook for an additional 5-7 minutes, until steak is cooked through and noodles are soft.
Add bean sprouts and bok choy to the pot, stir until they're cover with broth, and remove the pot from the heat. Let stand 5 minutes before serving.
To serve, spoon into bowls and top with the remaining green onions, shredded carrots, and an egg cut in half.