Heat 2 teaspoons of vegetable oil in a saucepan over medium heat. Add garlic and ginger; cook and stir until fragrant, about 30 seconds.

Stir in brown sugar, soy sauce, and water. Increase heat to medium-high; stir until sauce boils and slightly thickens, about 4 minutes. Remove sauce from the heat and set aside.

Place beef into a large bowl; add cornstarch and mix until beef is thoroughly coated. Set aside until most of the cornstarch has been absorbed, about 10 minutes.

Heat vegetable oil in a deep skillet to 375°F (190°C). Shake excess cornstarch from beef slices and drop into hot oil, a few at a time, stirring briefly and frying until edges become crisp, about 2 minutes per batch. Remove beef with a large slotted spoon; drain on paper towels.

Remove excess oil from the skillet, then heat the skillet over medium heat; add beef slices and stir in prepared sauce. Add green onions, bring to a boil, and cook until the onions have just softened and are bright green, about 1 to 2 minutes.