Mix the sauce: Whisk together the soy sauce, sesame oil, oyster sauce if using, and black pepper in a small bowl.
Scramble the eggs: Melt ½ tablespoon butter in a large skillet over medium heat. Scramble the eggs until fully cooked, then transfer to a plate.
Cook the vegetables: Add the remaining butter and vegetable oil. Cook the onion and garlic for 2–3 minutes. Add the frozen peas and carrots and cook for another 2–3 minutes.
Fry the rice: Add the chilled rice, breaking up any clumps. Pour the sauce over the rice and stir-fry for 4–5 minutes, until heated through.
Finish and serve: Return the eggs to the skillet, add the green onions, and toss for about 1 minute. Serve hot.