Freeze flank steak for 15-20 minutes, until just starting to firm. Cut against the grain into ⅛-inch thick slices, cutting long slices in half with the grain into two pieces.
In a bowl, toss steak slices with cornstarch. Add soy sauce, water, and rice wine, and toss to coat. Marinate for 15-20 minutes.
Make the Stir-Fry and Serve
In a bowl, combine low-sodium chicken broth, oyster sauce, light and dark soy sauces, sesame oil, brown sugar, cornstarch, and the white and light green parts of the scallions (about ½ cup scallions). Set aside.
Heat 1-½ teaspoons vegetable oil over high heat in a large (12-inch) skillet or wok until shimmering.
Pat the steak slices dry with paper towels and add half of them to the skillet, separating them with tongs into a single layer. Let them sit, undisturbed, for 1 minute. Then, use tongs to stir the beef around in the pan until it is brown around the edges, about 1 minute longer. Transfer to a bowl.
Add another 1-½ teaspoons oil to the pan, heat, and repeat the process with the remaining beef.
When all of the beef has been cooked and removed from the pan, heat 1 tablespoon of vegetable oil. Add the broccoli and cook, stirring, until crisp-tender, 2-4 minutes, depending on the size of your florets.
Clear an area in the center of the pan. Heat 1-½ teaspoons vegetable oil and add minced ginger and garlic. Cook for about 30 seconds, until fragrant, and then stir it into the broccoli. Add the meat and any accumulated juices back into the pan and toss to combine.
Whisk the chicken broth mixture to recombine any cornstarch that might've settled in the bowl. Pour the sauce around the pan. Lower heat to medium, and cook, stirring, until the sauce is thickened and evenly coats the meat and broccoli, about 1-2 minutes.
Season to taste with white pepper (if desired), sprinkle with remaining sliced scallions (dark green parts), and serve over rice.
Notes
*Due to the salt content in the oyster and soy sauces, it's important to use low-sodium chicken broth to make the stir-fry sauce. This is a relatively salty dish. To control the salt content further, you can use low-sodium soy sauce and season to taste with salt.