You don’t need to cook the eggs and rice separately—just add them in sequence and stir-fry. You can add carrots if you like. Onion is optional.
Make scrambled eggs in a pan with some oil, then set aside in a bowl. Season lightly with salt.
On low heat, add oil to the pan and stir-fry chopped green onion. Add rice and stir-fry together. Season with a little salt.
Add the scrambled eggs back in and mix. Adjust seasoning to taste if needed.
Turn off the heat, finish with a drizzle of sesame oil and a sprinkle of sesame seeds.