Preheat your oven to 375°F (190°C). Roll out the thawed puff pastry on a lightly floured surface into a 10-inch circle, slightly larger than your skillet, and place it in the refrigerator to stay cold.

In a 9-inch oven-proof skillet (preferably cast iron), melt the unsalted butter over medium heat. Sprinkle the granulated sugar evenly over the melted butter and stir gently to combine.

Arrange the apple halves tightly in the skillet, rounded side down, in a circular pattern. Cook over medium heat for 15 to 20 minutes, undisturbed, until the sugar and butter bubble and turn into a deep amber caramel.

Remove the skillet from the heat, carefully sprinkle the vanilla extract and salt over the apples, and let the mixture cool for 5 minutes to slightly thicken the caramel.

Drape the chilled puff pastry over the warm apples, carefully tucking the edges down the inside of the skillet to encapsulate the fruit. Prick the pastry a few times with a fork to allow steam to escape.

Bake in the preheated oven for 20 to 25 minutes, or until the puff pastry is puffed, deeply golden brown, and crisp.

Remove from the oven and let it rest for exactly 5 minutes (waiting too long will cause the caramel to stick). Place a large serving plate over the skillet, quickly and confidently invert the tart onto the plate, and serve warm.