In a small saucepan heat 2 teaspoons of the oil. Add the garlic and ginger, and saute for 30 seconds, then add the soy sauce and brown sugar and bring to a simmer, whisking frequently over medium heat. Allow it to simmer and reduce a bit, until it gets a glazey consistency, about 4 minutes. Set aside.
Meanwhile, toss the shaved steak in a mixing bowl with the cornstarch until evenly distributed.
Heat 2 tablespoons of oil in a large skillet (the bigger the better) over medium high heat. Line a counter or a large plate with paper towels. Sauté the beef in batches, taking care not to crowd the pan, and flipping it as it browns, about 2 minutes in all. Use tongs to remove the beef when it is browned, transferring it to the paper towels. Repeat until all of the beef is browned, adding more oil as needed (make sure the oil has a chance to get hot before you add the next batch of beef so it browns up nicely).
Pour off any additional oil and return the large skillet to medium high heat. Add all of the beef back to the pan, along with the sauce and scallions. Stir for about 3 minutes until the sauce is thickened and glazes all of the meat evenly. Serve over the rice.
What the Kids Can Do:
Slice scallions with supervision, measure the sauce ingredients, and help prepare the rice.