Pastry Dough
1. Make the pastry dough: in a food processor, pulse the flour and sugar together to combine. Add the butter and pulse until coarse crumbs form. Add the water, 1 tbsp at a time, pulsing just until the dough comes together. Transfer the dough to a lightly floured surface and gather it into a ball. Pat into a disk, wrap tightly in plastic wrap and refrigerate until chilled, at least 20 minutes.
2. On a lightly floured surface, roll out the dough into a 10", circle, about 1/4" thick. Transfer the dough to a lightly floured baking sheet, cover and refrigerate.
3. Preheat the oven to 400 degrees. Melt the butter in a 9" cast-iron skillet. Add the sugar and cook over moderately high heat, stirring, until a rich brown caramel forms, about 5 minutes. Remove from the heat. Pack the apple pieces in the skillet, rounded sides down but slightly leaning, in concentric circles.
4. Lay the pastry round on top of the apples, pressing gently to fit it in the pan. Bake the tart for about 35 minutes, or until the pastry is golden brown. Let cool slightly, then carefully invert the pan on a warm serving plate. Serve the tart warm or at least room temperature accompanied by the Cinnamon Ice Cream.
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