Prepare the mushrooms

1. Cut the stems off the morels and white mushrooms. Carefully wash the morels. To do this, soak them several times in a bowl filled with lukewarm water. Change the water until there is no more sand at the bottom of the basin. Drain in a strainer. Rinse the white mushrooms. Drain and quarter them.

Cook the mushrooms

1. Melt the butter in a sauté pan, then add the morels and white mushrooms. Season lightly with salt.
2. Add the wine, cover with a lid, and braise on a gentle simmer for 12 minutes. If the mushrooms dry out, add a few tablespoons water.

Prepare the eggs

1. Peel and mince the onion. Cut the cheese into a small dice.
2. Break the eggs into a deep earthenware bowl. Beat lightly and add the milk, onion, and cheese. Mix well, then add the cooked mushrooms. Mix again and season with a turn of the pepper mill.

Cook the omelet

1. Heat a little olive oil in a nonstick frying pan. Pour in the contents of the bowl and tilt the pan in all directions to spread it out well. While cooking the omelet, stir it and push the edges toward the center with a flexible spatula (scra-per). Cook the omelet until it is done to your liking. Slide it onto a plate and serve.