Warm a tablespoon of oil in a large deep-sided skillet over medium-high heat. Add the cabbage and the onion. Sprinkle with salt and pepper and toss to coat. Add about 2 tablespoons of water and cover with lid. Reduce heat to medium-low and cook covered for 3 minutes.
Uncover, stir, and continue cooking over high heat for 1-2 minutes more, while stirring frequently, until the vegetables soften and the cabbage edges begin to brown. Transfer to a bowl and set aside.
Warm a tablespoon of oil in a large non-stick skillet over medium heat. Swirl the oil to coat the surface of the pan. Pour the eggs into the pan and tilt as needed to spread across the skillet. Let the eggs cook for 2-3 minutes and then flip. Cook one more minute. Slide the egg out of the skillet onto a cutting board and allow it to cool. Cut into ½-inch wide strips and then cut into small pieces.
Return the skillet used for the vegetables to the stove. Add the remaining 2 tablespoons of oil to the pan and set the heat to medium-high. Add the cold rice and stir to coat. Continue cooking, while stirring frequently, for about 2 minutes until the rice begins to crisp a bit.
Add the soy sauce and sesame oil. Stir well, to distribute the sauce throughout the rice. Cook for 5 minutes, stirring frequently until the rice is heated through.
Add the cooked vegetables, the eggs, and optional meat if desired, to the skillet with the rice and continue cooking for an additional minute until everything has warmed through. Taste and adjust the salt and pepper, as desired. Stir in the green onions before serving.