Preheat your oven to 375°F (190°C).
Peel, core, and quarter the apples. Set aside.
In a 9- or 10-inch ovenproof skillet or cast-iron pan, melt the butter over medium heat.
Add the sugar and cook, stirring constantly, until the mixture turns a deep golden brown and caramelizes. Be careful not to burn it.
Remove the skillet from the heat and stir in the vanilla extract.
Carefully arrange the apple quarters in the caramel in a circular pattern, placing them tightly together. They will shrink as they cook, so pack them in tightly.
Return the skillet to medium heat and cook the apples for about 10-15 minutes, basting them with the caramel occasionally, until they start to soften.
On a lightly floured surface, roll out the puff pastry sheet to a size slightly larger than your skillet.
Carefully place the puff pastry over the apples, tucking the edges down around the apples inside the skillet.
Place the skillet in the preheated oven and bake for 25-30 minutes, or until the pastry is golden brown and puffed.
Remove the skillet from the oven and let it cool for about 5 minutes.
Carefully run a knife around the edges of the skillet to loosen the tart.
Place a large serving plate over the skillet and, using oven mitts, carefully invert the skillet and plate together to release the tart onto the plate.
Enjoy your meal!