Preheat your oven to 400°F (200°C). Use an oven-safe, 9-inch skillet (cast iron is ideal) for this recipe.

Prepare the caramel: Sprinkle the sugar evenly over the bottom of the skillet. Place the skillet over medium heat and allow the sugar to melt without stirring. Swirl the pan occasionally until the sugar turns a deep amber color (about 5-8 minutes).

Remove the skillet from the heat immediately. Carefully dot the melted caramel with the cubes of butter, allowing them to melt slightly into the sauce. Arrange the peach halves snugly in the skillet, cut-side down, over the caramel layer.

Cut the thawed puff pastry to fit the skillet, slightly larger than the diameter. Place the pastry sheet over the peaches, tucking the edges down between the peaches and the side of the pan to fully enclose the fruit.

Bake for 30 to 35 minutes, or until the pastry is golden brown and puffed up. Remove from the oven and let it rest on a wire rack for exactly 10 minutes.

To unmold, place a serving plate upside-down over the skillet. Carefully and swiftly invert the skillet and plate together. If any peaches stick, gently replace them onto the tart using a fork. Brush lightly with vanilla extract if desired.