Whisk eggs and milk, season with salt, pepper and nutmeg. Grease a flat dish. Pour in the egg milk, cover and allow to set in a hot water bath for 30 to 40 minutes.
In the meantime, clean, wash and drain the mushrooms. Heat the oil in a pot and sauté the mushrooms. Deglaze with broth and bring to the boil once. Season to taste with salt and pepper. Turn out the eggs and cut into triangles.
Wash and finely chop the parsley. Arrange the eggs on a plate with the broth. Sprinkle with parsley.