Peel and cut the onion. In a medium skillet, heat oil or butter until hot, add onion and sauté for about 3 minutes. Add sliced mushrooms and sauté for another 7-8 minutes. Season with salt and pepper. Remove from heat.
Preheat oven to 375 degrees F. Lay the puff pastry on a parchment paper and transfer it onto a baking sheet. Prick all over with a fork.
Evenly spoon the mushrooms mix on top.
In a bowl, whisk together eggs and cream. Season with salt and pepper. Pour over the mushrooms.
Place the cheese slices all over the pie. Sprinkle with nutmeg.
Bake on the bottom rack for about 35 minutes, until golden.