Prepare the Apples:

1. Peel and Slice the Apples: Peel, core, and cut the apples into 1/4-inch thick slices. Toss them in lemon juice to prevent browning.

Make the Caramel:

2. Melt Butter: In a 9-inch oven-safe skillet (preferably cast iron), melt the butter over medium heat.
3. Add Sugar: Once the butter is melted, add the brown sugar, stirring constantly until the sugar dissolves and begins to bubble. Continue to cook until the caramel thickens and turns a rich golden brown (about 5 minutes).
4. Add Rosemary: Add the rosemary sprigs to the caramel and let them infuse the syrup for 1-2 minutes.

Arrange the Apples:

5. Layer the Apples: Carefully arrange the apple slices in a circular pattern in the skillet, making sure to tightly pack them. The apples will shrink as they cook, so it’s okay to layer them high.
6. Cook the Apples: Let the apples cook in the caramel for about 15 minutes. Stir occasionally to ensure even caramelization.

Add the Puff Pastry:

7. Prepare the Puff Pastry: Roll out the puff pastry to fit the size of your skillet. Once the apples are cooked and caramelized, carefully drape the puff pastry over the apples, tucking the edges down around the sides of the skillet.
8. Bake the Tarte Tatin: Transfer the skillet to the oven and bake at 375°F (190°C) for about 25-30 minutes, or until the pastry is golden and puffed.

Finish and Serve:

9. Cool and Invert: Let the tarte cool for about 5 minutes, then carefully invert it onto a serving plate. Remove the rosemary sprigs and garnish with fresh rosemary leaves.
10. Serve: Serve warm, with a dollop of whipped cream or a scoop of vanilla ice cream if desired.