Preheat your oven to 375°F. Use a 9-inch round baking pan or oven-proof skillet and set it aside.

In the pan or skillet, melt the butter over medium heat. Add the brown sugar, granulated sugar, cinnamon (if using), and a pinch of salt. Stir until the sugars dissolve, and the mixture begins to bubble.

Arrange the apple slices tightly in a circular pattern in the pan. Cook over medium heat for about 10 minutes, or until the apples are tender and caramelized. Stir gently as needed to prevent burning. Remove the pan from the heat and let it cool slightly.

On a lightly floured surface, roll out the puff pastry and cut it into six even rectangles.

Place a sheet of puff pastry on top of each portion of caramelized apples, tucking the edges down slightly with a fork to secure the pastry.

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

Remove the pan from the oven and allow the tarts to cool for a few minutes. Carefully flip each tart onto a plate by placing a plate over the pan and turning it upside down. Use caution, as the caramel will be very hot.