Making a successful tarte tatin is all about well-cooked pastry and rich, thick caramel sauce. The juices from the peaches will thin and lighten the caramel when it’s in the oven, so make sure it’s cooked long enough and is dark brown -- but not burnt -- before topping with the pastry. Don’t be afraid to bake the pastry until it's very crisp and browned. It can take a while to fully cook through!
Preheat the oven to 375˚ F. Melt the butter in a medium ovenproof nonstick skillet over medium-high heat. Stir in the sugar and salt until it looks like wet sand. Reduce the heat to medium and add the peaches. Cook, shaking the skillet, until the syrup is dark brown, 10 to 15 minutes.
Meanwhile, trim the corners from the puff pastry to make a round. Remove the skillet from the heat and top the peaches with the puff pastry, tucking in the edges; cut three 2-inch slits into the top.
Bake until browned on top, about 30 minutes. Let cool 10 minutes in the skillet, then invert onto a rimmed serving plate.