In a mixing bowl, combine the sliced beef with 2 tablespoons of soy sauce and 1 tablespoon of cornstarch. Toss to coat and let it marinate for about 10 minutes.
Heat 2 tablespoons of vegetable oil in a large skillet or wok over medium-high heat. Add the marinated beef and stir-fry until browned, about 3-4 minutes. Remove the beef from the skillet and set aside.
In the same skillet, add the remaining 1 tablespoon of oil. Add the minced garlic and ginger, stir-frying for about 30 seconds until fragrant.
Add the broccoli florets to the skillet and stir-fry for 2-3 minutes until they are bright green and slightly tender.
Return the beef to the skillet. In a separate bowl, mix the remaining soy sauce, oyster sauce, beef broth, and sesame oil. Pour this sauce over the beef and broccoli mixture.
Cook for an additional 2-3 minutes, stirring constantly until the sauce thickens and everything is heated through. Season with salt and pepper to taste.
Remove from heat and serve immediately over cooked rice, if desired.