1. Warm 2 Tbsp. oil in a wok or large nonstick skillet over medium-high heat until oil is shimmering. Add cabbage, onion and white parts of scallions and stir-fry for 30 seconds. Add ginger and garlic to skillet, toss to combine with cabbage mixture and stir-fry for an additional 15 to 20 seconds. Transfer to a wide, shallow bowl or plate.
2. Warm remaining 3 Tbsp. oil in same skillet. Add rice, mix to coat with oil and spread out in pan (but don't press flat). Let rice cook undisturbed for 2 minutes, then stir. Let cook undisturbed 1 minute longer, then push to edges of pan. Add eggs to center of pan and stir to scramble. Break eggs into pieces and stir into rice.
3. Drizzle soy sauce over rice, add chicken and stir to combine well. Cook for 1 to 2 minutes longer to warm through. Stir in cabbage mixture and half of scallion greens and continue cooking and stirring until warmed through, about 1 minute. Sprinkle with remaining scallion greens and serve.