Preheat the oven to 180°C (350°F).
In a heavy ovenproof skillet, melt the butter over medium heat. Add the sugar and cook until it turns a golden caramel color.
Arrange the apple quarters snugly in the caramel, with rounded sides facing down.
Cook the apples for 10 minutes, then remove from heat.
Top the apples with the puff pastry, tucking the edges inside the pan.
Bake in the preheated oven for 30 minutes, until the pastry is golden and crisp.
Let the tart cool for a few minutes, then carefully invert it onto a serving plate.