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1. FOR THE CRUST: Pulse flour, sugar, and salt in food processor until combined. Add shortening and process until mixture has texture of coarse sand, about ten 1-second pulses. Scatter butter pieces over flour mixture and process until mixture is pale yellow and resembles coarse crumbs, with butter bits no larger than small peas, about ten 1-second pulses. Transfer mixture to medium bowl.
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2. Sprinkle 3 tablespoons ice water over mixture. With blade of rubber spatula, use folding motion to mix. Press down on dough with broad side of spatula until dough sticks together, adding up to 1 tablespoon more ice water if dough does not come together. Turn dough out onto sheet of plastic wrap and flatten into 4-inch disk. Wrap dough and refrigerate 30 minutes, or up to 2 days, before rolling out. (If dough is refrigerated longer than 1 hour, let stand at room temperature until malleable.).
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3. FOR THE FILLING: Adjust oven rack to upper-middle position (between 7 and 9 inches from heating element) and heat oven to 500 degrees. Whisk cider, syrup, lemon juice, cornstarch, and cinnamon (if using) together in medium bowl until smooth. Heat butter in 12-inch heatproof skillet over medium-high heat. When foaming subsides, add apples and cook, stirring 2 or 3 times until apples begin to caramelize, about 5 minutes. (Do not fully cook apples.) Remove pan from heat, add cider mixture, and gently stir until apples are well coated. Set aside to cool slightly.
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4. TO ASSEMBLE AND BAKE: Roll out dough on lightly floured work surface, or between 2 large sheets of plastic wrap, to 11-inch circle. Roll dough loosely around rolling pin and unroll over apple filling. Brush dough with egg white and sprinkle with sugar. With sharp knife, gently cut dough into 6 pieces by making 1 vertical cut followed by 2 evenly spaced horizontal cuts (perpendicular to first cut). Bake until apples are tender and crust is a deep golden brown, about 20 minutes. Let cool 15 minutes; serve.
