Heat oil; mix eggs and soy sauce: In a wok or large nonstick skillet, heat 1 tablespoon oil over high. Meanwhile, in a small bowl, lightly beat eggs with ¼ teaspoon soy sauce to combine.
Cook eggs; cut into small pieces: Add eggs to wok and swirl to coat bottom of pan. Cook, without stirring, until almost set, 1 minute, then fold in thirds with a spatula. Transfer cooked eggs to a work surface and cut into ½-inch pieces.
Cook garlic, ginger, and scallion whites: Add 1 tablespoon oil to wok and swirl to coat. Add garlic, ginger, and scallion whites and cook, stirring constantly, until fragrant, 30 seconds.
Add pork and cook; stir in carrots, peas, and rice: Add pork and cook, breaking up meat with a wooden spoon, until just cooked through, about 3 minutes. Add carrots, peas, and rice and stir to combine.
Add egg, soy sauce, and vinegar: Add cooked egg, 2 tablespoons soy sauce, and vinegar and cook, stirring constantly, until rice is coated, about 1 minute. Let cook, undisturbed, until warm, about 1 minute. Top with scallion greens and serve.
