Heat 2 tablespoons vegetable oil in a small saucepan over medium-low heat. Add the garlic and ginger and sauté until golden, about 2 minutes. Add the soy sauce and water, stirring to combine.
Stir in the brown sugar and increase the heat to medium. Bring the sauce to a boil for 3 minutes. Remove the sauce from the heat and set it aside.
Slice the flank steak against the grain into 1/4-inch pieces, then toss it with the cornstarch. Place the coated pieces of steak in a sieve and shake off any excess cornstarch. Allow the steak to sit 10 for minutes.
Line a plate with paper towels.
Place a large skillet or wok over medium-high heat and add 1/3 cup vegetable oil. Once the oil is hot (but not smoking), add the beef to the pan and sauté until it is seared on all sides but barely cooked in the center, about 2 minutes. Remove the steak from the skillet with a slotted spoon and transfer it onto a paper towel-lined plate. Pour any excess oil out of the skillet.
Place the skillet back over medium heat. Add the prepared sauce to the hot pan (it should come to a boil almost immediately), then add the reserved steak and cook at a boil, stirring constantly, until the sauce has thickened to the consistency of syrup, 3 to 5 minutes. Add the sliced scallions and stir to combine. Serve.