In a small bowl, combine egg white, cornstarch, Chinese cooking wine and 1 tablespoon oil. Whisk together until smooth and frothy. Add beef and massage marinade onto meat. Marinate in the refrigerator for about 30 minutes. In a colander, drain beef.
In a bowl, combine stir-fry sauce ingredients: water, soy sauce, Hoisin sauce, Chinese cooking wine and sugar. Stir until well blended. Add cornstarch and stir until dissolved and mixture is smooth. Set aside.
In a wok over high heat, heat about 1/4 cup oil. Add beef in a single layer and cook for about 2 to 3 minutes or until lightly browned. With a slotted spoon, remove meat and drain on paper towels.
Remove excess oil except for about 1 tablespoon. Add garlic and ginger and cook, stirring regularly, for about 30 seconds or until aromatic. Add onions and cook, stirring regularly, for about 30 to 40 seconds or until half done.
Stir stir-fry sauce to disperse cornstarch that may have settled on the bottom and add to wok. Cook, whisking regularly, until smooth and thickened.
Return beef to wok and add green onions. Continue to cook, tossing gently, for about 1 to 2 minutes or until meat is heated through. Serve hot.