01 - Heat vegetable oil in a large skillet over medium heat. Add diced chicken and cook until lightly browned and just cooked through. Add onion and carrot; sauté until softened, about 2-3 minutes. Stir in peas, then add the cooked rice, breaking up any clumps. Season with ketchup and soy sauce, mixing thoroughly until rice is evenly coated. Taste and adjust seasoning with salt and pepper as needed. Divide fried rice into two portions and shape each into an oval mound. Place each mound on a serving plate.
02 - In a bowl, whisk eggs with milk and a pinch of salt until well combined. Heat butter in a nonstick skillet over medium heat. Pour in half the egg mixture, swirling to coat the pan evenly. As the eggs just begin to set but are still slightly runny on top, carefully slide one portion of fried rice onto one side of the omelet. Using a spatula, gently fold the other side of the omelet over the rice, forming a half-moon shape. Carefully transfer to a serving plate, seam side down. Repeat for the second portion.
03 - Drizzle ketchup decoratively across the omelet. Sprinkle with parsley, if desired. Serve immediately while hot.