Preheat the oven to 350 degrees F (175 degrees C).

Heat a 10-inch cast iron frying pan over medium heat. Melt sugar, butter, rosemary, and cinnamon together in the hot pan, stirring constantly, until syrup is a rich caramel color, about 8 minutes.

Reduce heat to low. Add apple slices, arranging them in a decorative swirl starting from the outside edges of the pan, rounded sides-down, as tart will be turned upside-down when served. Simmer, uncovered, until apples are slightly tender yet firm, about 10 minutes. Shake the pan occasionally to prevent scorching. Remove from heat.

Trim corners of puff pastry to form a 10-inch circle. Place pastry over apples and push down between apples and the edge of the pan using the tip of a knife.

Bake in the preheated oven until pastry is golden and puffed, about 10 minutes.

Remove from the oven and let cool for 10 minutes.

Loosen the edges of the tart from the pan using a knife. Invert a serving plate over the pan and then, holding the pan and plate together firmly, invert them together. Lift off the pan. Cut tart into wedges and serve warm or at room temperature.