Place the chilled apple halves round side down in the skillet. Try to keep gaps between the apples to a minimum. Place skillet in the fridge.
Preheat the oven to 425 with a rack in the center.
Remove the rough puff or store bought pastry from the fridge and let sit for 5 minutes. Unwrap the dough and place on a floured work surface.
Beat the dough with a rolling pin to help make it more pliable. Dust the dough with flour as needed and roll out to a 12-inch round shape. Set an 11-inch plate over the top of the dough and cut around it to create an 11-inch round piece of pastry dough. Using the tines of a fork, prick all over the pastry round.
Drape the pastry over the apples and using your hands or a large spoon, tuck the edges of the dough down between the apples and the sides of the skillet. Place in the fridge for 15 minutes to chill.
Bake the tart for 20 minutes.
Reduce the temperature of the oven to 350 degrees Fahrenheit and bake until the pastry is puffy and golden brown all over and the caramel is bubbling up around the sides of the skillet.
Let the tarte rest in the skillet for 10 minutes.
Over a sink, invert the tart onto a wire cooling rack. While the tart is still warm, use a pastry brush to generously brush the tarte with the syrup made earlier. Slide the tarte onto a plate and serve warm.