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1
Center a rack in oven and preheat oven to 375 degrees. Choose a 9- or 10-inch ovenproof skillet - we like cast iron - or, if you've got one, a tarte Tatin pan. Line a small baking sheet with parchment or a silicone mat.
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2
Working on a floured surface, roll puff pastry out until it is about 1/8 inch thick (it can be thicker, if you'd like). Using a paring knife, cut dough into a circle that is 1-inch larger than diameter of the pan you're using. Prick dough all over with tines of a fork and transfer it to baking sheet. Cover dough and refrigerate it while you work on apples.
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3
Melt butter in skillet over medium heat. Tilt pan so that the sides have a thin coating of melted butter. Sprinkle sugar over butter and remove from heat.
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4
Fit a layer of apples into skillet, putting apples into pan rounded side down and making concentric circles. It's important to pack in the apples - they will shrink as they cook, so you want to make sure they are snug in the pan. When you've got a tight single layer, cut remaining apple quarters in half (if you have some left) and scatter them over first layer.
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5
Put pan over medium heat and cook until sugar turns a deep caramel color. It will likely take about 15 minutes, more or less, but you may have to lower heat after a bit as to not burn sugar. Remove from heat. Drizzle pistachio oil over cooked apples. Remove pastry from fridge and center it over fruit, loosely tucking in any overhang (it's okay if you have a double layer of dough around edges), or not - the oven's heat will shrink pastry to size. Rinse baking sheet and place the tart pan on it so that the tart does not bubble up and drip in the oven.
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6
Bake for 30 to 40 minutes, or until pastry is baked through and puffed.
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7
To Serve: Turn pan upside down over a large, rimmed serving plate and pop out tart.
