Preheat your oven to 400°F (200°C) and ensure your puff pastry sheet is thawed but remains cold in the refrigerator until ready to use.

In a 9-inch ovenproof skillet, spread the softened butter evenly across the bottom, then sprinkle the sugar, vanilla extract, cinnamon, and salt in an even layer over the butter.

Arrange the apple halves tightly in the skillet, curved side down, nesting them closely together as they will shrink slightly during the cooking process.

Place the skillet over medium heat on the stovetop and cook for 15 to 20 minutes, allowing the butter and sugar to melt and bubble into a rich, golden-brown caramel.

Remove the skillet from the heat and carefully drape the cold puff pastry sheet over the caramelized apples, tucking the excess edges down into the sides of the pan.

Cut three small slits in the center of the pastry with a sharp knife to allow steam to escape while baking.

Bake for 20 to 25 minutes, or until the pastry is puffed, crisp, and deep golden brown on top.

Remove from the oven and let it cool in the pan for 5 to 10 minutes; do not let it cool completely, or the caramel will stick to the pan.

Place a large, flat serving platter face down over the skillet, then quickly and carefully invert the skillet to release the tart onto the platter.

Reposition any apples that may have shifted, and serve the tart warm with a dollop of crème fraîche or a scoop of vanilla ice cream.