Make the sauce: In a bowl, whisk soy sauce, water or stock, brown sugar, hoisin (if using), rice vinegar, sesame oil, and cornstarch until smooth.
Set aside.
Heat the pan: Use a large skillet or wok over medium-high heat. Add 1 tablespoon oil and swirl to coat.
Brown the beef: Add the beef mince and break it up. Let it sear for 2–3 minutes before stirring, then cook until browned with some crispy edges.
Season lightly with a pinch of salt and pepper. Transfer to a plate, leaving a little fat in the pan.
Cook aromatics: Add the remaining oil if needed. Sauté onions for 2 minutes until softened.
Add garlic and ginger, and cook 30–45 seconds until fragrant. Add chilies if using.
Add vegetables: Toss in bell pepper and the white parts of the scallions. Stir fry 2–3 minutes until crisp-tender.
Return beef to pan: Add the cooked mince back in and toss to combine with the vegetables.
Pour in sauce: Give the sauce a quick stir, then add it to the pan.
Stir constantly for 1–2 minutes as it thickens and glazes the beef. If it gets too thick, splash in a tablespoon of water.
Finish: Stir in the green parts of the scallions. Taste and adjust—add a pinch more brown sugar for sweetness, soy for salt, or vinegar for brightness.
Serve: Spoon over hot rice or noodles, or pile into lettuce cups.
Garnish with sesame seeds and a squeeze of lime if you like.