01 - Heat vegetable oil in a large skillet over medium heat. Add diced chicken and cook until lightly browned and just cooked through, about 3-4 minutes. Add onion and carrot, sautéing until softened, approximately 2-3 minutes. Stir in frozen peas and cook for 1 minute.
02 - Add the cooked rice to the skillet, breaking up any clumps with a wooden spoon. Pour in ketchup and soy sauce, mixing thoroughly until the rice is evenly coated with a reddish hue. Taste and adjust seasoning with salt and pepper as needed. Divide the fried rice into two equal portions and shape each into an oval mound. Place each mound on individual serving plates.
03 - In a mixing bowl, whisk together eggs, milk, and a pinch of salt until the mixture is uniform and slightly frothy. Ensure the whites and yolks are fully incorporated.
04 - Heat half the butter in a nonstick skillet over medium heat. Pour in half the egg mixture, swirling the pan to create an even circular layer. As the eggs just begin to set but remain slightly runny on top, carefully slide one portion of fried rice onto one side of the omelet.
05 - Using a spatula, gently fold the opposite side of the omelet over the rice, creating a half-moon shape. Carefully invert the skillet to transfer the omelet onto the prepared plate, positioning it seam side down over the rice mound. Repeat the process with the remaining butter, egg mixture, and rice portion.
06 - Drizzle ketchup in a decorative pattern across the top of each omelet. Sprinkle with finely chopped fresh parsley if using. Serve immediately while the omelet remains silky and the rice is hot.