In a bowl, combine beef, 1 tablespoon soy sauce, and 1 tablespoon Shaoxing wine and toss to coat. Place in refrigerator and let marinate for at least 20 minutes and up to 3 hours.

Meanwhile, combine remaining soy sauce with cornstarch and stir with a fork to form a slurry. Add remaining Shaoxing wine, chicken stock, oyster sauce, sugar, and sesame oil. Set aside. Combine garlic, ginger, and scallion whites in a bowl and set aside.

When ready to cook, heat 1 tablespoon oil in a wok over high heat until smoking. Add half of beef and cook without moving until well seared, about 1 minute. Continue cooking while stirring and tossing until lightly cooked but still pink in spots, about 1 minute. Transfer to a large bowl. Repeat with 1 more tablespoon of oil and remaining beef, adding beef to same bowl. Wipe out wok.

Add 1 tablespoon oil to wok and heat over high heat. Add half of broccoli and cook until crisp-tender and lightly charred, about 1 minute. Transfer to bowl with beef. Repeat with remaining oil and remaining broccoli. Return wok to high heat until smoking. Return beef and broccoli to wok and add garlic/ginger/scallion mixture. Cook, tossing and stirring until fragrant, about 30 seconds. Add sauce and scallion greens and cook, tossing and stirring constantly until lightly thickened, about 45 seconds longer. Transfer to a serving platter.