Preheat your oven to 375°F (190°C).
Use a round baking pan or oven-proof skillet approximately 9 inches in diameter.
In your pan or skillet, melt the butter over medium heat. Add both types of sugar, cinnamon and a pinch of salt, stirring until the sugars dissolve and the mixture starts to bubble.
Add the apple wedges to the pan, arranging them tightly in a circular pattern. Cook over medium heat for about 10 minutes, until the apples are somewhat tender and caramelized. Be careful not to burn the caramel.
While the apples are caramelizing, roll out your puff pastry on a lightly floured surface. Cut in 6 even rectangles.
Once the apples are caramelized, let them cool off for a few minutes.
Then place 1 sheet of butter on the baking sheet, add 4 slices of caramelized apple and the puff pastry on top. Then push the edges down using a fork.
Bake in the preheated oven for 30 minutes, or until the puff pastry is golden and puffed.
Let the tart cool for a few minutes after removing it from the oven. Then carefully flip the tart onto the plate. Be very careful, as the caramel and pan will be very hot.
Dust some confectioners sugar over the tarts.
Serve warm or at room temperature, ideally with a dollop of whipped cream or a scoop of vanilla ice cream.