In a bowl, combine the sliced beef with 1 tablespoon of soy sauce, garlic, ginger, and salt and pepper. Let the beef marinate for about 10 minutes.
While the beef is marinating, blanch the broccoli in boiling water for about 2-3 minutes until vibrant green and slightly tender. Drain and set aside.
Heat the vegetable oil in a skillet or wok over medium-high heat. Once hot, add the marinated beef and stir-fry for 3-4 minutes until browned and cooked through. Remove the beef from the skillet and set it aside.
In the same skillet, add the blanched broccoli and stir-fry for an additional 2 minutes.
Return the beef to the skillet. In a small bowl, mix together the remaining soy sauce, oyster sauce, cornstarch, and water to create the sauce. Pour this mixture over the beef and broccoli, stirring to combine.
Cook for another 2-3 minutes, allowing the sauce to thicken and coat the beef and broccoli well.
Taste and adjust seasoning with additional salt and pepper if needed. Serve hot over cooked rice if desired.