Position an oven rack in the middle of the oven and preheat to 400 degrees F.
Put the puff pastry on a cutting board, put an inverted 10-inch nonstick ovenproof skillet on top and cut the pastry with a knife. Transfer the pastry round to a parchment-lined baking sheet and poke all over with a fork. Refrigerate until ready to use.
Put the sugar in the bottom of the 10-inch skillet. Heat over medium heat, shaking the skillet frequently, until the sugar is a dark caramel color, about 5 minutes. Let cool 5 minutes.
Meanwhile, separate the necks and bulbs of the squash using a sharp knife. Reserve the bulbs for another use. Cut the stems off the necks and peel the necks with a peeler. Using a mandoline, slice the squash as thinly as possible while keeping them as whole circles.
Start to arrange 1 layer of overlapping squash slices over the bottom of the skillet in concentric circles. Sprinkle with half the thyme, half the sage and 1/2 teaspoon salt. Lay down another layer of squash and top with the remaining thyme and sage and another 1/2 teaspoon salt. Top with any remaining squash and place the puff pastry round on top.
Transfer the skillet to a baking sheet and bake until the pastry is golden brown, about 45 minutes. Let cool for 3 minutes in the skillet and invert onto a plate. Top with freshly grated Parmesan.
Copyright 2019 Television Food Network, G.P. All rights reserved.