Position an oven rack 4 to 6 inches below the heating element of your broiler and preheat to high or 450°F. Arrange the Hatch Chiles in a single layer on an aluminum foil-lined baking sheet.

Place the chiles under the broiler and roast until the skin is charred and blistered, about 5 minutes. Using tongs, flip the chiles over and broil on the other side until the skin is charred and blistered, about 5 minutes.

Place chiles in a paper sack or plastic food-storage bag, close it tight and let the chiles steam for 20 minutes. Take the chiles out of the bag and rub off the skin. Remove seeds and stems and dice.

Heat oven to 400°F. Toss the squash in a roasting pan with the garlic. Drizzle with 2 tablespoons oil and season well. Roast for 30 minutes or until soft and starting to brown. Flip squash over after 15 minutes. Remove and squeeze the garlic out of its skin, mash and gently mix with the squash, set aside to cool.

Heat remaining oil in a small pan and fry the onion until soft and lightly golden. Add the mushroom and fry for 3-4 minutes more. Remove from heat and set aside.

In a large bowl stir in goat cheese, ground Hatch Chile pepper, Hatch Chiles, onions, mushrooms and squash. Adjust seasoning with salt and pepper. Set aside.

Position an oven rack in the lower third of the oven and heat the oven to 400°F.

Line a large rimmed baking sheet with parchment. In a small bowl, make an egg wash by beating one egg with 1 tablespoon of water.

Unfold the puff pastry dough on a floured surface. With a floured rolling pin, roll the dough into a 12x14-inch rectangle. Cut the rectangle in half lengthwise to form two 6x14-inch rectangles. Place one of the dough rectangles onto the parchment-lined baking sheet. Brush with beaten egg, pierce all over with a fork, and bake for 10 minutes until lightly golden and slightly risen. (Save the remaining egg wash.)

Lightly dust the remaining piece of puff pastry with flour and then gently fold it in half lengthwise; don’t crease the fold. Using a sharp knife, cut 1-1/2-inch-long slashes at 1-inch intervals along the folded side of the dough; leave at least a 1-inch border on the remaining three sides.

Pile the filling onto the partly-cooked base leaving a 1 inch border. Brush the border with a little beaten egg and carefully lay the uncooked pastry atop the filling on one side, then unfold it to completely cover the filling, pressing down the edges to seal. Glaze top with a little more egg and bake for 30- 35 minutes until the pastry is golden, risen and cooked through.

Cut into slices and serve.