To begin, preheat your oven to 190°C (375°F). Peel, core, and quarter the apples. Set them aside in a large bowl. This will allow them to stay fresh while you prepare the caramel.

In an oven safe skillet (about 9-10 inches in diameter), melt the butter over medium heat. Once melted, sprinkle the sugar evenly over the butter and cook until the sugar dissolves and begins to caramelize, turning a light amber colour. Stir occasionally, but be careful not to let the caramel burn. This should take about 5-7 minutes.

Once the caramel reaches a deep golden colour, carefully arrange the apple quarters in the skillet, making sure to place them tightly together. Let the apples cook in the caramel for 10 minutes, gently turning them once halfway through to coat them evenly in the caramel. This step enhances the flavour and texture of the apples.

Remove the skillet from heat. Add the vanilla extract and a pinch of salt to the caramelized apples, swirling the skillet slightly to distribute the flavour. Be cautious as the vanilla may cause the caramel to bubble.

On a lightly floured surface, roll out the puff pastry slightly to ensure it will cover the entire skillet. Carefully drape the pastry over the apples, tucking the edges down around the apples inside the skillet. This ensures the pastry will cradle the apples and absorb the caramel.

Transfer the skillet to the preheated oven and bake for 25-30 minutes, or until the puff pastry is golden brown and crisp. The pastry should puff up slightly as it bakes, creating a beautiful texture that contrasts with the soft, caramelized apples.

Once baked, remove the skillet from the oven and let it rest for 5 minutes. Carefully run a knife around the edges of the skillet to loosen the pastry. To flip the tart, place a large serving plate over the skillet, then quickly and carefully invert the skillet onto the plate. The caramelized apples should now be on top.

Serve the Tarte Tatin warm, either on its own or with a scoop of vanilla ice cream or a dollop of crème fraîche. The contrast between the warm caramelized apples and the buttery pastry with the cold cream enhances the eating experience. Garnish with a few mint leaves or a dusting of powdered sugar for a final elegant touch.