Heat oven to 400F, line a baking sheet with parchment paper.
In a large pan, saute mushrooms, onion, and garlic in olive oil, until the liquid comes out and mushrooms are very soft. Add wine, continue to cook until all wine has boiled off. Stir in thyme, season with salt and pepper to taste. Remove from heat.
Unfold thawed pastry sheet onto parchment lined pan. Roll out a little to elongate it slightly in both directions.
Use a sharp knife to make cut 1″ wide strips (like a fringe) along the two long edges, ending ⅓ of the way towards the center.
Evenly spread filling over the center ⅓ of the dough, sprinkle with goat cheese.
Take one of the end strips and fold it across the filling, at a slight diagonal. Take the top edge strip from the other side, and fold it across the filling, ending just over top of the first one. Continue, alternating sides, until the whole thing is “braided”.
Whisk together egg and 1 tablespoon balsamic vinegar, use a pastry brush to liberally spread it over the whole braid.
Bake for 30 minutes, or until puffy and golden brown.
Remove from heat, drizzle with more balsamic vinegar if you like. Serve hot!