Heat ½ tablespoon of the vegetable oil in a wok (or large nonstick frying pan) over medium heat. Add beaten egg and gently scramble with a rubber spatula until just softly set, 1-2 minutes. Transfer to a bowl and set aside, returning the wok to the stovetop.
Add 1 tablespoon of oil to the wok, then add onion, garlic, salt, and pepper to the wok. Cook for 4-5 minutes until the onion is tender, then push it to the side of the wok.
Add the remaining oil and let heat for a few seconds. Add cold cooked rice, frozen peas and carrots, and soy sauce. Mix and stir-fry 3-4 minutes until the rice is heated through.
Remove from the heat and stir in scrambled eggs, green onions, and sesame oil. Serve immediately.