Step 1 In a large skillet, melt the butter over medium-high heat. Add the shallot and cook, stirring frequently, until aromatic, 1 minute.
Step 2 Stir in the mushrooms and continue to cook, stirring frequently, until the mushrooms give off their moisture and begin to color, 3 to 5 minutes. Continue cooking, stirring occasionally, until almost all of the moisture evaporates and the mushrooms color to a rich brown, 8 to 10 minutes.
Step 3Remove the pan from the heat and add the wine. Return the skillet to the heat and stir, scraping any flavoring from the bottom of the pan, until the wine is absorbed, 2 to 3 minutes. Remove from heat, stir in the thyme and season with one-fourth teaspoon salt and one-fourth teaspoon of pepper, or to taste.
Step 4Set the mushroom mixture aside until cool, then stir in the grated cheese.
Step 5Roll the puff pastry into a rectangle measuring 20 by 10 inches; the dough will be just over one-eighth-inch thick. Using a ruler, divide the dough into 8 (5-inch) squares. Then place the squares on 2 parchment-lined baking sheets. Refrigerate for 15 to 20 minutes.
Step 6Meanwhile, divide the filling into 8 portions: Gently press some of the filling into a one-third-cup measure. Unmold the filling and, using your hands, press the filling into an oval ball (like a small football). Repeat with the remaining filling; you should use all of the filling. Set aside.
Step 7Brush the beaten egg along 2 of the 4 edges of each square. To assemble the turnover, place one mound of filling in the center of each square. Working quickly, pick up a square and fold it in half diagonally and press gently, sealing the edges together. Continue forming the remaining turnovers, then check each turnover again to make sure the seal holds. Brush the remaining beaten egg over each of the turnovers, and prick the top of each with a knife to form vent holes.
Step 8Refrigerate the turnovers on the baking sheets for 15 to 20 minutes before baking. Meanwhile, heat the oven to 425 degrees. Bake the turnovers, 1 sheet at a time on the center rack, until puffed and golden brown, about 30 minutes, rotating the sheet halfway through. Serve the turnovers warm or at room temperature.