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Toss the sliced beef with 1 tablespoon of soy sauce and let sit for 10 to 15 minutes while you prep everything else.
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In a small bowl, whisk together the remaining soy sauce, brown sugar, rice vinegar, sesame oil, red pepper flakes, cornstarch, and beef broth. Set aside.
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Heat 1.5 tablespoons of vegetable oil in a large skillet or wok over high heat. When the oil is shimmering, add half the beef in a single layer and cook without moving for 2 to 3 minutes until browned. Flip and cook another minute. Remove to a plate and repeat with remaining beef and oil.
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Add a bit more oil if needed, then add the broccoli florets. Stir-fry for 5 to 6 minutes until the broccoli is tender-crisp and starting to char at the edges.
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Add the minced garlic and ginger to the pan and cook for about 30 seconds until fragrant.
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Return the beef to the pan and pour in the sauce. Toss everything together and cook for 2 to 3 minutes until the sauce thickens and everything is heated through.
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Taste and adjust seasoning with salt and pepper. Garnish with sliced green onions and sesame seeds. Serve immediately over steamed rice.