Heat the oven to 375°F (190°C) and set out a round baking pan or oven-safe skillet that is about 9 inches in diameter.

In the pan or skillet, melt the butter over medium heat. Add both sugars, the cinnamon, and a pinch of salt, stirring until the sugars are fully dissolved and the mixture starts to bubble. Arrange the apple wedges tightly in a circular pattern and cook over medium heat for about 10 minutes, until the apples are slightly tender and caramelized. Watch closely so the caramel does not burn.

While the apples are cooking, roll out the puff pastry on a lightly floured surface and cut it into 6 even rectangles. Once the apples are done, let them cool for a few minutes.

Place 1 sheet of butter on the baking sheet, top it with 4 slices of caramelized apple, and lay the puff pastry over the apples. Gently press the edges down with a fork.

Bake in the preheated oven for 30 minutes, or until the puff pastry is puffed and golden brown.

Remove from the oven and let the tart cool for a few minutes. Carefully flip it onto a plate, being cautious since the pan and caramel will be very hot.

Lightly dust the tarts with confectioners sugar and serve warm or at room temperature, ideally with whipped cream or a scoop of vanilla ice cream.