In a large measuring cup or a bowl with a lip, whisk together egg, green onion, ham, carrot, salt and sesame oil until well combined.
Then in a 8 inch round non-stick pan set over medium heat, lightly grease your pan with cooking oil. Once the pan is hot enough, whisk your egg batter again and then pour JUST ENOUGH egg batter until a thin layer forms. You may need to swirl your pan a bit or use chopsticks to evenly disperse the ingredients. LOWER heat to very low. Let this fry for 2-3 minutes.
Once the thin layer forms an edge and the layer is mostly solid with several wet looking areas (your batter should not be runny at all), it’s time to begin rolling your omelette. With a spatula, first loosen the edges and the bottom a bit. Then fold the omelette 1 inch inward and continue rolling until you reach halfway and stop. Turn up the heat to medium again.
Then loosen the opposing edge and carefully pull the rolled omelette back to the halfway point of your pan.
Next pour just enough batter aiming for the edge until you fill that side of the pan. Do not fill the other side of the pan beyond the rolled part.
Allow that layer to fry for 2-3 minutes and bring heat down to VERY LOW.
Repeat the above process until you have depleted your batter. Once your omelette is completely rolled, let it sit on the pan for 5 minutes over low heat to cook through.
Finally transfer to a cutting board to slice into 0.75-inch-thick pieces. Serve and enjoy!