Preheat oven to 425°F Position the rack to the middle of the oven.
Heat the oil and butter in a 12 saute' pan over medium heat. Once the butter starts to foam, add the mushrooms and garlic.
The mushrooms will quickly absorb all the fat in the pan.
Sprinkle with 1/2 t salt and stir until the mushrooms release their moisture and start to shrink, about 2-3 minutes, maybe a bit longer.
Increase the heat so that you hear a steady sizzle; stir occasionally.
In about 5 minutes, when the liquid evaporates, and the mushrooms start to brown, give an occasional sweep of the spoon, once a minute, to let the mushrooms edges brown nicely, about 2-4 more minutes. REsist the inclination to stir to often.
Turn off the heat and stir in the pepper and parsley. Add more salt if needed. Add the other herbs and set aside to cool.
Thaw the puff pastry sheet at room temperature until pliable, 30-45 minutes.
Lightly dust a work surface with flour. Unfold the pastry sheet and roll it into a rectangle about 10x15 inches and 1/8 thick.
Slide onto a baking sheet lined with parchment paper.
Prick the pastry crust all over with a fork but NOT the outer 1 on all sides.
Over the 1 border, brush with the beaten egg wash. You will not use all of it.
Bake the pastry until it begins to puff up and surface feels dry, about 5 minutes.
Prick the crust with a fork or skewer to release the steam if it puffs up like a pillow.
Scatter the mushrooms onto the pastry, leaving a 1 border.
Bake until the border is puffed up and golden brown, about 10 minutes.
Remove from oven and scatter the cheese on top the mushrooms and continue baking until the cheese melts, about 2-3 minutes.
Cool briefly and slice. Using a pizza cutter/wheel will help wit tis.