Preheat your oven to 400°F (200°C).

1. In a 9-inch ovenproof skillet or Tarte Tatin pan, melt the butter over medium heat. Add the sugar and stir until the sugar dissolves and the mixture bubbles. Let it cook for 5-7 minutes until the caramel turns golden brown.
2. Add the peach halves to the pan, cut side down. Cook for about 10 minutes, allowing the peaches to soften and caramelize. If desired, sprinkle cinnamon for added warmth.
3. Remove the skillet from heat and drizzle the vanilla extract over the peaches.
4. Roll out the puff pastry on a lightly floured surface and place it over the caramelized peaches. Tuck the edges of the dough around the peaches to seal it.
5. Transfer the skillet to the preheated oven and bake for 25-30 minutes or until the puff pastry is golden brown and crisp.
6. Remove the skillet from the oven and let it cool for a few minutes. To flip the tart, place a large plate over the skillet and gently invert the pan. Let it sit for a moment before lifting the pan off to reveal the caramelized peaches.
7. Garnish with fresh mint leaves if desired, and serve warm with whipped cream or vanilla ice cream.