Put eggs and milk in a bowl and mix well.
Finely chop onion, carrot, mushroom, and ham.
Heat cooking oil in a pan and stir-fry the prepared vegetables and ham.
Add rice to make fried rice.
Shape the fried rice into a round mound.
Heat cooking oil in a pan and thinly cook the egg mixture.
Place the fried rice on the egg sheet, wrap it, and shape it.
Serve the omurice on a plate and drizzle with sauce to finish.