In a medium bowl, combine the soy sauce, oyster sauce, cornstarch, beef broth, brown sugar, and black pepper. Stir well to dissolve the cornstarch and set aside.
Heat 1 tablespoon of vegetable oil in a large skillet or wok over medium-high heat.
Add the sliced beef to the skillet and stir-fry for 2-3 minutes until browned but not fully cooked. Remove the beef from the skillet and set aside.
Add the remaining 2 tablespoons of vegetable oil to the skillet.
Add the minced garlic and grated ginger to the skillet and stir-fry for 30 seconds until fragrant.
Add the broccoli florets and sliced carrots to the skillet. Stir-fry for 4-5 minutes until the vegetables are tender-crisp.
Return the beef to the skillet with the vegetables.
Pour the sauce mixture over the beef and vegetables. Stir well to coat everything evenly.
Cook for an additional 2-3 minutes, stirring frequently, until the sauce has thickened and the beef is cooked through.
Drizzle the sesame oil over the stir fry and give it a final stir.
Remove from heat and serve immediately with steamed rice or noodles.