Slice the beef thin. Cut against the grain into 1/4-inch slices. This keeps it tender and quick-cooking.
Mix the sauce. In a bowl, whisk soy sauce, oyster sauce, hoisin (or sugar), rice vinegar, sesame oil, broth, cornstarch, and optional chili. Set aside.
Marinate briefly. Toss half the sauce with the beef and let sit 10 minutes while you prep veggies.
Keep the other half to finish the stir-fry.
Preheat your pan. Heat a large skillet or wok over high heat until very hot. Add 1 tablespoon oil and swirl to coat.
Sear the beef in batches. Spread beef in a single layer. Cook 60–90 seconds per side until browned but still tender.
Remove to a plate. Don’t overcrowd.
Stir-fry the vegetables. Add another tablespoon oil. Stir-fry harder veggies (carrots, broccoli) for 2 minutes, then add quicker-cooking ones (peppers, snap peas, onions) for 2–3 minutes more.
Add aromatics. Push veggies aside, add a little oil if needed, then add garlic and ginger.
Cook 30 seconds until fragrant.
Combine and sauce. Return beef and any juices to the pan. Pour in the remaining sauce. Toss 1–2 minutes until glossy and slightly thickened.
Season and finish. Taste and adjust with a splash of soy for salt, vinegar for brightness, or chili for heat.
Sprinkle green onions and sesame seeds.
Serve over rice. Fluff warmed rice. Spoon stir-fry on top and finish with a squeeze of lime if you like.