Cut the beef into 2cm strips.

Slice the onion and mushrooms thinly. Trim the shimeji mushrooms and separate them into small clusters.

Heat butter in a pan over medium heat and sauté the onion until softened. Add the beef and mushrooms and cook until browned.

Add sugar and flour to the pan and stir briefly. Pour in red wine and bring to a boil. Add the sauce ingredients and simmer on low heat for 20 minutes. Season with salt and pepper.

In a bowl, whisk eggs with milk and mayonnaise.

Heat oil in a pan over medium heat. Pour half of the egg mixture into the pan and cook until semi-set. Repeat to make two omelettes.