Take puff pastry out of freezer and allow to thaw while you prepare other ingredients.
Heat a skillet over medium heat, add butter and olive oil. When skillet is hot and butter is melted, add the onion with a pinch of salt and sauté until onions have started to soften, about 5 minutes.
Add the garlic and mushrooms with a bit more salt, cook for about a minute until the pan starts to get dry, about a minute.
Add the balsamic vinegar and wine. Cook until no liquid remains, 5-7 minutes.
Add the fresh herbs and cook for another 30 seconds to a minute. Remove from heat.
Taste and adjust salt and pepper as needed.
Roll out the puff pastry just a bit so that it is a rectangle, about 9 and ½ by 11 inches.
Score a 1 inch border around the edge with a sharp knife. You want to cut almost all the way through, but not quite. Prick the inside of the border with a fork.
Spread the grated cheese over the inner square of dough, trying not to go over the scored edge.
Top the cheese with the mushroom mixture.
Brush the border with the scrambled egg.
Bake in the oven for about 15 minutes or until the crust is golden brown and puffed up.
Allow to cool for 10 minutes before cutting and serving.
Notes
This would serve 6-8 as an appetizer and 4 as a meal.