Preheat the oven to 375°F (190°C).
Melt 1 tablespoon of the butter in a skillet over medium heat. Stir in the brown sugar, granulated sugar, cinnamon, and salt. Cook until the sugars dissolve and the mixture begins to bubble.
Add the apple slices to the caramel and cook over medium-low heat for about 10 minutes, until softened and caramelized, being careful not to break them up or burn the caramel.
Roll out the puff pastry on a lightly floured surface and cut it into 6 rectangles.
Cut the remaining 3 tablespoons of butter into 6 equal half-tablespoon pieces. On a parchment-lined baking sheet, place 1 butter piece, top with about 4 slices of caramelized apple, and cover with a rectangle of puff pastry. Press the edges down with a fork to seal. Repeat with the remaining butter, apples, and pastry to make 6 tarts.
Bake the tarts for 30 minutes, or until the puff pastry is puffed and golden brown.
Cool for a few minutes, then carefully flip the tarts onto a plate. Be cautious — the caramel will be very hot.
Dust the tarts with powdered sugar.
Serve the tarts warm or at room temperature, with whipped cream or a scoop of vanilla ice cream if desired.