Melt the sugar in an ovenproof frying pan over medium heat until it turns into a golden caramel. Try not to stir it too much; if necessary, gently swirl the pan.
Carefully add the butter and stir until melted. Mix in the vanilla bean paste.
Arrange the apple quarters tightly within the caramel, with the rounded sides facing down.
Place the puff pastry over the apples, tucking the edges down around the fruit.
Make a small slit in the centre to allow steam to escape.
Brush the pastry with the beaten egg and bake for 25–30 minutes, until golden and crisp.
Allow the tart to rest for 5 minutes, then carefully turn it out onto a serving plate while the caramel is still warm.