1 small butternut squash, peeled, seeded, and cut into 1-inch cubes
1 tablespoon olive oil
sea salt and freshly ground black pepper
6 tablespoons (¾ stick) unsalted butter, plus more if needed
8 ounces mixed fresh mushrooms, such as oysters, chanterelles, shitakes, or hedgehogs, cleaned and thinly sliced (3½ to 4 cups)
2 shallots, minces
¼ cup all-purpose flour
3½ cups low-sodium chicken broth
4 cups cooked shredded chicken
1 tablespoon chopped fresh sage or 1 teaspoon dried crumbled sage
1 sheet frozen puff pastry, such as Pepperidge Farm or Dufour Pastry Kitchens, thawed in the refrigerator
1 large egg, lightly beaten with 2 tablespoons of milk, for egg wash
Preheat the over to 400°F. Lightly grease an ovenproof 9- or 10- inch skillet or a 2-quart baking or soufflé dish.
Toss the squash in a bowl with the olive oil to lightly coat. Season with salt and pepper to taste, spread evenly on a rimmed baking sheet, and bake until tender and golden brown, 25 to 30 minutes. Remove from the oven and set aside.
Melt the butter in a large skillet over medium-high heat and cook the mushrooms and shallots, stirring frequently, for 5 to 6 minutes, until light bown, adding more butter if needed. Add the flour and cook, stirring constantly, until light brown, 3 to 4 minutes longer. Slowly stir in the broth, scraping the brown bits from the bottom of the pan, and bring to a low boil, stirring constantly, until slightly thick, about 5 minutes.
Add the chicken and sage, season with salt and pepper to taste, and stir to mix. Reduce the heat to a simmer and cook, stirring frequently, until the mixture is thick and creamy, about 10 minutes. Remove from the heat, stir in the reserved squash, and allow to cool.
Meanwhile, roll the puff pastry on a lightly floured surface just to even out the lines, cut the pastry to fit the top of the baking dish, leaving enough excess to allow the pastry to drape over the sides by ¼ inch.
Fill the prepared skillet or baking dish with the chicken mixture, leaving about ½ inch of space at the top, and place the pastry over the dish. Cut several slits in the crust to allow steam to escape. Decorate as desired, brush with the egg wash, and refrigerate for at least 1 hour.
Preheat the oven to 375°F. Place the pie on a rimmed baking sheet and bake until the filling is bubbling around the edges and the pastry is puffy and golden brown, 45 to 50 minutes. Remove from the oven and serve warm.