In a medium bowl or large zip-lock bag, combine the beef, soy sauce, brown sugar, garlic, ginger and onions. Refrigerate for at least an hour, or overnight.
Heat a drizzle of oil in a large skillet set over medium-high heat, remove the beef from the marinade and stir-fry, tossing with tongs, just until cooked through. If you like, chop the meat into smaller pieces to make it easier to wrap.
Fill soft lettuce leaves with steamed rice, beef and kimchi, and wrap to eat.