Preheat oven to 375F.
Lay out a large piece of waxed or parchment paper on a flat surface and very lightly flour it. Lay the 2 pieces of cold puff pastry on the paper, overlapping the edges on one side by 2 inches. Gently press the overlapped area together with your fingertips to seal it and make one large piece.
Turn your skillet upside down and very gently lay it in the center of the pastry (you don’t want the skillet to push down on the pastry too much). Measure out 2 inches from the edge of the skillet and cut a circle out of the pastry; throw away the excess. Remove the skillet, roll up the pastry with the paper and stash it in the refrigerator to keep it chilled.
Peel and cut the apples in half. Scoop out the core using a small melon-baller, if you have one, or a teaspoon.
Put the butter, sugar, salt, ginger and cinnamon in the bottom of a 10-inch iron skillet. Place the skillet on the stove over low heat and stir gently until the syrup ia completely melted and mixed.
Lay the vanilla pod halves on top of the syrup in the shape of a cross. Place the apple halves in the skillet, flat side down. If you can’t fit all of the halves into the skillet, and there is space left, cut the remaining halves into chunks and gently stuff them into the spaces. Stud the apple pieces evenly with the cloves.
Remove the puff pastry from the refrigerator and carefully unroll it over the apples. Tuck the edges of the pastry underneath the apple halves all around the skillet. Make several 1-inch slits in the pastry with a sharp knife. Increase the heat to medium and cook for about 15 minutes shaking the pan and rotating it regularly.
When syrup forms in the base of the pan and it turns light amber, dip a pastry brush into it and paint some evenly over the puff pastry. Make sure to cover the entire surface of the pastry to encourage even browning.
Place the skillet in the preheated oven. Bake until the pastry is a deep golden color, about 20-25 minutes.
Remove the skillet from the oven and let it stand for 5 minutes. Shake the skillet gently and place a large, flat plate on top. Fold a tea towel into each hand to protect yourself from the skillet’s heat and the hot caramel. Grab the skillet and plate together and then quickly and carefully flip the tart onto the plate. Cut into pieces and serve hot.