Begin by slicing the flank steak into ½-inch thick strips. This ensures that the beef cooks quickly and remains tender. For best results, slice against the grain. Once sliced, toss the beef in cornstarch until evenly coated.
In a mixing bowl, combine the soy sauce, brown sugar, water, rice vinegar, minced garlic, and minced ginger. Whisk these ingredients together until the sugar is dissolved. Set aside a small bowl to prepare the cornstarch slurry by mixing 1 tablespoon of cornstarch with 2 tablespoons of water.
Heat the vegetable oil and sesame oil in your skillet over medium-high heat. Once hot, add the beef strips in a single layer. Cook for about 2-3 minutes on each side, or until the beef is browned and cooked through. Remove the beef from the skillet and set it aside.
In the same skillet, pour the prepared teriyaki sauce mixture. Bring it to a simmer and allow it to cook for 2-3 minutes. Then, stir in the cornstarch slurry and cook for an additional minute or until the sauce has thickened.
Return the cooked beef to the skillet and toss it in the teriyaki sauce, ensuring each piece is well-coated. Allow it to heat through for another minute. Serve over steamed rice, noodles, or alongside stir-fried vegetables.