Preheat the oven to 425 degrees F. On a floured work surface, roll the puff pastry into a 12-inch square, about 1/8 inch thick. Invert a 10-inch ovenproof nonstick skillet over the pastry. Using the skillet as a guide, cut a 10-inch round of pastry. Transfer to a plate and chill in the refrigerator while you prepare the apples.
Heat the same skillet over medium heat. Melt the butter and then sprinkle over the sugar and add the apples on top. Once the sugar begins to melt, adjust the heat so the juices bubble rapidly. Cook, stirring, until the apples are tender and juices are light golden brown, 10 to 12 minutes. Remove from the heat.
Using a fork, nudge the apples into 2 concentric circles (it doesn't have to be perfect) in the skillet. Top with the chilled pastry round and tuck the edges in. Gently prick the pastry in 4 or 5 places with a fork. Bake for 10 minutes at 425 degrees F, until the pastry is puffed and light golden. Reduce the oven temperature to 375 degrees F and bake until the pastry is a deep golden brown, 10 to 12 minutes more. Remove from the oven and let sit 10 minutes.
To unmold, place a serving plate over the skillet and quickly but carefully flip the pan. The tart will unmold onto the plate, and if a few apple slices stick to the pan, return them to the tart. Let cool at least 20 minutes before serving to allow the syrup to sink into the tart. Serve slightly warm or at room temperature.