Heat oil in a Dutch oven or soup pot over high heat. Add beef and cook, stirring often, until just cooked, about 2 minutes. Transfer to a plate using tongs, leaving the juices in the pot.
Add bok choy to the pot and cook, stirring, until wilted, about 2 minutes. Add broth and water, cover and bring to a boil. Add noodles and soy sauce; simmer until the noodles are soft, about 4 minutes. Return the beef to the pot and cook until heated through, 1 to 2 minutes more. Ladle into bowls and sprinkle with bean sprouts and basil. Serve hot.
If you have a little extra time before dinner, put the flank steak in the freezer for about 20 minutes to help make it easier to slice thinly.