In a small bowl, whisk together the soy sauce, sesame oil, brown sugar, minced ginger, garlic, and cornstarch until the sugar and starch are dissolved.
Heat 1 tablespoon of vegetable oil in a large skillet or wok over high heat until shimmering.
Add the beef slices in a single layer and sear without moving for 1 minute, then toss and cook for another 1-2 minutes until browned. Remove beef from the pan and set aside.
Add the remaining 1 tablespoon of oil to the same pan. Add the broccoli florets and sliced bell pepper.
Stir-fry the vegetables for 3-4 minutes, adding a tablespoon of water if needed to steam the broccoli slightly until tender-crisp.
Return the cooked beef to the pan and pour the sauce mixture over the ingredients.
Toss continuously for 1-2 minutes until the sauce has thickened and coated the beef and vegetables thoroughly.
Remove from heat and serve immediately.