Heat a wok or large skillet over medium-high heat.
In a small bowl, lightly whisk together the 2 large eggs, ⅛ teaspoon salt, and ¼ teaspoon ground black pepper.
Add 1 tablespoon of butter into the heated wok or skillet. Once the butter melts, add in the eggs and scramble until they are no longer runny but still a soft scramble. Carefully remove the cooked eggs from the skillet or wok back into the small bowl. Set aside.
Add another 1 tablespoon of butter to the heated wok or skillet. Once the butter melts, add in the 1 small chopped onion and move around in the pan until the onion is lightly coated with the butter. Allow the onion to continue to cook until it becomes translucent.
Add in the remaining 2 tablespoons of butter into the wok or skillet along with the cooked onion. Once it melts add in the 4 cups cooked rice.
Add in the 2 tablespoons soy sauce and 1 teaspoon toasted sesame oil with the rice. Stir the rice frequently, breaking it up as needed. Once the fried rice has been heated thoroughly and has also lightly browned, add in the egg and stir to evenly distribute.
Serve warm with some yum yum sauce and store any leftover in an airtight container in the refrigerator up to 3 days.