Crack the eggs into a bowl and whisk them thoroughly until there are no lumps of egg white.
Heat butter in a small pan over medium-low heat. Once the butter starts to melt, use your wrist to swirl the pan so that the butter coats the entire surface, including the sides. Once the butter has completely melted and the pan is evenly coated, pour in the beaten eggs.
Hold chopsticks in your dominant hand and quickly stir the egg mixture, as if you were making very fine scrambled eggs. The eggs will start to set at the edges, so be sure to keep pushing the solidified parts toward the center. With your non-dominant hand, continue to shake the pan back and forth.
When the eggs are partially cooked and the bottom part doesn’t move much when you shake the pan, stop stirring for about 10 seconds to let the bottom cook.
Remove the pan from the heat and place it on a damp cloth to lower the pan's temperature. Turn off the heat for now.
Place the ketchup fried rice in the center of the omelet, shaping it into a long, horizontal mound. Be sure to leave some space at both the front and back of the rice, as well as a little on the sides.
Use a spatula to lift the far side of the omelet and fold it over the rice. Once it is covered, gently press the rice against the egg you just folded over.
Shake the pan to flip the rice and the back half of the omelet over the front half. Then, use a spatula to gently shape the omelet.
Return the pan to the stove and cook the seam side over medium-low heat for about 10 seconds.
Serve on a plate and top with ketchup.