Slice steak into thin strips. Toss with 1 tablespoon soy sauce and let stand for 20 minutes.
Heat a large, heavy skillet or a wok over high heat. Add oil and swirl to coat pan evenly. Add scallions, ginger and garlic; stir-fry until fragrant.
In batches, add beef and stir-fry until browned, about 2 minutes. Return beef to pan and add in peppers and snow peas. Stir-fry for about 1 minute. Stir in remaining soy sauce and oyster sauce. Stir-fry for 1 minute more.