To prepare the mushroom sauce, heat butter in a large skillet over a medium-high heat. Once the butter begins to sizzle, add the onions and saute for about 1 minute, then add the mushrooms and continue cooking until both are caramelized and reached deep golden brown color, about 2-3 minutes.
Carefully pour in the wine, followed the heavy cream. Deglaze the pan by scraping brown bits from the bottom and the sides of the pan. Add a pinch of fresh dill or other herbs that you like. Bring to a gentle boil and cook until the sauce has thickened. Season with salt and pepper to taste. Remove the pan from heat and set aside.
In a small bowl, beat three eggs with a fork or a whisk until just combined. The eggs should not be frothy. Mix in a little bit of fresh dill.
Heat butter in a non-stick pan over medium-high heat. Once the butter is sizzling, spread mushroom slices around the pan. Let them cook for about 20 seconds, then pour the beaten eggs evenly around the pan. Season with a pinch of salt and pepper.
Let the eggs cook for 10 seconds, then left one side of the egg cake and tilt the pan down toward the lifted side to let the uncooked egg run down and fill the gap. Repeat with the other sides, or until there is no runny egg on top and even doneness is achieved.
Spread shredded cheese along the center of the egg omelette. Top with 2/3 of the mushroom sauce, reserving 1/3 of the sauce for garnish. Quickly fold and transfer to a serving platter.
Top the omelette with the remaining 1/3 of the mushroom sauce, the sprinkle some freshly chopped herbs, like dill and green onions.
This omelette is big and can easily feed two, even three people.