Preheat the oven to 375 degrees.
In a large skillet, heat 1 tablespoon of olive oil and the butter. Saute the onion 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 sage 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..
In a large bowl, toss the squash with the remaining tablespoon 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 onions and mushrooms and toss.
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 onions 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-8-10 inch rectangle about 1/4 inch thick. Cut the pastry into six 4-inch squares and repeat with the second sheet
Spoon 2 tablespoons of the shiitake-squash mixture onto each square and top with 1 tablespoon of the Humble Goat plain 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.
Brush any remaining egg wash over the tops of the turnovers and bake until golden brown, approximately 15 minutes.
Serve alongside a ramekin of The Salad Girl's Curry Fig Vinaigrette as a dipping sauce