1.Line a large baking sheet with parchment paper and set aside. In a bowl, mix together goat cheese, olive oil, garlic and thyme. Set aside.
2.In a large pan over medium-high heat, melt butter, then add mushrooms and shallot. Do not overcrowd the pan. Season with 1/2 teaspoon salt and 1/4 teaspoon pepper.
3.Cook until the mushrooms release all their liquid and most of it boils away, about 5 minutes. Flip halfway through. Transfer to a plate to cool slightly. Preheat oven to 375° F.
4.Roll puff pastry dough out into a 10-inch by 15-inch rectangle on the parchment paper. Score the edges by running a sharp knife through the top layer of puff pastry. Leave a 1-inch border, like a picture frame, and do not cut all the way through the dough. Prick the dough on the inside with a fork all around, not the border/edge.
5.Spread goat cheese mixture evenly over the puff pastry inside the scored border. If the mixture is difficult to spread, drop in spoonfuls and use the back of the spoon to smear it. Spoon mushroom mixture on top.
6.Make the egg wash by whisking together egg with water. Use a pastry brush to brush edges of the tart. Bake at 375° F for 20 to 25 minutes or until the pastry is puffed and golden. Cool for 15 minutes before cutting and serving. Use fresh thyme sprigs for garnish.