01 - Heat vegetable oil in a large skillet over medium heat. Add the cubed chicken and cook, stirring occasionally, until no longer pink throughout, approximately 3 to 4 minutes.
02 - Add the finely chopped onion and minced garlic to the skillet. Sauté until the onion becomes translucent and fragrant, about 2 minutes. Stir in the frozen mixed vegetables and continue cooking for an additional 2 minutes until warmed through.
03 - Add the cold cooked rice to the skillet, breaking apart any clumps. Stir-fry vigorously until the rice is evenly heated and well combined with the other ingredients. Season with soy sauce, ketchup, salt, and pepper, stirring until the rice is uniformly coated and takes on an even amber color throughout.
04 - Remove the skillet from heat. Divide the fried rice into two equal portions and shape each into a tight oval mound on a plate or cutting board. Set aside while preparing the omelettes.
05 - Crack the eggs into a mixing bowl, add milk and salt, then whisk vigorously until fully blended and slightly frothy.
06 - Melt half the butter in a nonstick skillet over medium-low heat, tilting to coat the surface evenly. Pour in half of the egg mixture and swirl the pan to spread it into a thin, even layer. Cook gently, stirring lightly with chopsticks in small circular motions, until the edges are set but the center remains slightly soft and runny. Carefully place one oval mound of fried rice onto one half of the partially set omelette. Using a spatula, fold the omelette over the rice to enclose it completely. Gently slide the wrapped omurice onto a serving plate, seam side down.
07 - Wipe the skillet clean if needed, add the remaining butter, and repeat the omelette process with the remaining egg mixture and second rice mound.
08 - Drizzle additional ketchup over each omurice in a zigzag or decorative pattern. Serve immediately while hot.