Cut the chicken into bite-sized pieces. Heat oil in a skillet and cook the chicken skin-side down. Flip, add the sake, cover, and cook through.

Dice the carrot and onion. Blanch the diced carrot in boiling water since it takes longer to cook.

Add the cooked chicken and vegetables to the skillet, season with salt and pepper, and sauté. Add the rice and continue to stir-fry. Add ketchup and mix to make ketchup rice.

Crack 2 eggs into a bowl, add 1 tablespoon milk and a pinch of salt, and beat well. Make an omelet as you normally would for omurice.

For the demi-glace sauce: Toast the flour in a skillet until golden brown, then set aside.

Wipe the skillet lightly, melt the butter, and sauté the sliced onion and shimeji mushrooms until soft. Remove from heat and stir in the toasted flour.

Add the water, chicken bouillon, Worcestershire sauce, and ketchup to the skillet. Simmer for about 10 minutes, stirring occasionally.

Pour plenty of demi-glace sauce over the omurice and serve!