Warm the vegetable oil in a large skillet or wok over medium-high heat, letting it get hot enough to start shimmering if using a wok.
Pour in the whisked eggs and quickly scramble them, stirring constantly until they start to form soft curds.
When the eggs are just set but still moist, add the chilled rice, breaking up any clumps with your spatula.
Push the rice and eggs to one side of the pan; add the remaining oil in the free space, then stir in the chopped green onions.
Drizzle the soy sauce and sesame oil around the edges of the pan, then mix everything together until the rice is evenly coated and slightly browned.
Remove from heat and serve the egg fried rice immediately while warm.