Step 1: Heat 1 tablespoon butter in a large skillet over medium heat. Add diced chicken and cook until browned, about 5 minutes.
Step 2: Add diced onion and carrot to the skillet. Cook until vegetables soften, about 3-4 minutes.
Step 3: Add cooked rice to the skillet, breaking up any clumps. Stir to combine with chicken and vegetables.
Step 4: Add ketchup and soy sauce to the rice mixture. Stir well until rice is evenly coated and heated through, about 3 minutes. Season with salt and pepper to taste. Divide into 2 portions and set aside.
Step 5: In a bowl, whisk together eggs, milk, and a pinch of salt until well combined.
Step 6: Heat a non-stick skillet over medium-low heat. Add remaining butter and pour half the egg mixture into the pan, swirling to create a thin, even layer.
Step 7: Cook the egg until set but still slightly runny on top, about 2-3 minutes. Place one portion of rice in the center of the omelet.
Step 8: Carefully fold the edges of the omelet over the rice to create a neat package. Slide onto a plate.
Step 9: Repeat steps 6-8 with remaining egg mixture and rice portion.
Step 10: Garnish with additional ketchup and chopped parsley before serving.