Preheat the grill to medium temperature.
Shake cream eggs (if used) in a small bowl.
Melt the butter in a heat-resistant pan with a diameter of 20 cm with a non-stick coating on a medium high heat. Put the mushrooms and fry until golden brown, about 4 minutes.
Slightly lower the heat, pour in the egg mixture and immediately mix in the grated cheese, pushing the set edges of the omelet from the sides of the pan. Salt, pepper and sprinkle with parsley.
Cook on the stove for a couple of minutes, without stirring, to make the omelet brown from below. Place it under the grill for about 2 minutes, it will rise and lightly browned on top.
Put an omelet on a plate and serve immediately, adding arugula, pre-sprinkled with oil.