You don't need to cook the eggs and rice separately; you can add them one after the other in the pan. Carrots can be added if you like. Onion is not recommended.
Make scrambled eggs in an oiled pan and set them aside in a bowl. Season lightly with salt.
On low heat, add oil to the pan, sauté green onions, then add rice and stir-fry. Season with salt.
Add the scrambled eggs back in, mix well, and adjust seasoning if needed.
Turn off the heat, finish with a drizzle of sesame oil and a sprinkle of sesame seeds.