Slice the beef thinly across the grain into bite-sized strips. Toss the beef with 1 tablespoon soy sauce and 1/2 tablespoon cornstarch in a bowl; let it sit while you prep the vegetables (about 8–10 minutes).
Whisk together the remaining soy sauce, oyster sauce, water or broth, remaining cornstarch, and brown sugar in a small bowl until smooth; set aside. This will be the sauce that finishes the stir fry.
Heat a large wok or heavy skillet over high heat until very hot. Add 1 tablespoon neutral oil and swirl to coat. Working in one or two quick batches, spread half the beef in the pan and let it sear without moving for 30–45 seconds, then stir-fry until just browned (about 1–2 minutes). Transfer cooked beef to a plate and repeat with remaining oil and beef.
Reduce heat to medium-high and add the remaining 1 tablespoon oil. Add the sliced onion and stir for 1 minute, then add garlic and ginger and cook until fragrant (about 30 seconds), stirring constantly to prevent burning.
Toss in the shredded cabbage and white parts of the green onions. Stir-fry, stirring frequently, until the cabbage softens but still has some crunch, about 3–4 minutes. If the pan looks dry, splash in 1–2 tablespoons of water to help the cabbage steam and prevent scorching.
Return the beef and any accumulated juices to the pan. Give the prepared sauce a quick stir and pour it into the wok. Stir continuously until the sauce thickens and evenly coats the beef and cabbage, about 1 minute.
Turn off the heat, drizzle the toasted sesame oil over the stir fry, and fold in the green parts of the scallions and red pepper flakes (if using). Let the pan sit for 1–2 minutes so flavors meld before serving.
Taste and adjust with additional soy sauce or a pinch of black pepper if needed. Serve hot over steamed rice or noodles, garnished with extra sliced green onions if desired.