How to Make It
Preheat the oven to 375°. In a large skillet, heat 2 tablespoons of the olive oil. Add the diced leeks and cook over moderate heat, stirring occasionally, until softened, about 5 minutes. Add the shiitake and cook, stirring often, until their liquid has evaporated, about 5 minutes. Stir in the garlic and the thyme and cook for 2 minutes. Season with salt and pepper and transfer to a medium bowl.
Line a large rimmed baking sheet with parchment paper and lightly oil the parchment. In a large bowl, toss the squash with the remaining 2 tablespoons of olive oil and season with salt and pepper. Spread the squash on the prepared baking sheet and bake for about 25 minutes, until softened and starting to brown. Add the squash to the leeks and mushrooms and toss.
Line a large rimmed baking sheet with parchment paper. On a lightly floured work surface, gently roll out the puff pastry to a 12-by-16-inch rectangle about 1/4 inch thick. Cut the pastry into twelve 4-inch squares. Spoon 2 tablespoons of the shiitake-squash mixture onto each square and top with 2 tablespoons of the goat cheese. Lightly brush the edge of the squares with some of the beaten egg. Fold the squares over to form triangles and crimp the edges decoratively with a fork.
Arrange the turnovers on the prepared baking sheet about 1/2 inch apart. Brush the tops of the turnovers with the remaining beaten egg. Bake for about 25 minutes, until the pastry is golden brown. Serve warm or at room temperature.