Preheat the oven to 392°F.
Pour 1/2 cup of sugar into the bottom of a Tatin dish. Caramelize it over high heat with the juice of half a lemon, then remove from heat.
Peel, core, and quarter the apples. Place them in the dish.
Dot the apples with small pieces of butter and cover with the puff pastry.
Prick the pastry with a fork before baking for 20 minutes.
Prepare the sauce: Scrape the seeds from the split vanilla bean into the heavy cream.
Pour 3/4 cup of sugar into a heavy-bottomed saucepan. Heat until you get a brown caramel.
Add the cream, mix, bring back to a boil, then let cool. Refrigerate.
Remove the tart from the oven and wait 10 minutes before turning it out onto a serving plate: place the plate upside down on the tart and flip quickly.
Serve immediately with the vanilla caramel sauce.