When the onion softens, add the rice and corn. Stir-fry briefly, season with a little salt, then add the ketchup and stir-fry. Transfer to a plate.

Crack the eggs into a bowl, season with salt and pepper, add the milk, and mix well.

Heat butter in the same frying pan, pour in the eggs all at once, and quickly stir until just set but still soft. Place the eggs over the ketchup rice.

Make the demi-glace sauce: Thinly slice the onion, cut the tomato into 1/2-inch (1 cm) cubes, and separate the shimeji mushrooms into small clusters.

In the same frying pan, add 1 cup (240 ml) water, the sliced onion, and shimeji mushrooms. Bring to a simmer and cook until the onion is translucent. Add the tomato and simmer a little longer.

Add 1 piece of Hayashi rice roux and stir until dissolved. Finish by adding 1 teaspoon milk and simmer for 1 minute. Pour plenty of sauce over the omurice!