APPLE PIE, CIRCUS-STYLE

Steps:

• Make the dough: Put the bread flour, whole-wheat flour, sugar and salt in the bowl of a food processor, and whir to combine. Drop in the chunks of butter, then process in 8 to 10 long pulses, scraping the bowl often, until the ingredients look like coarse crumbs and hold together when pressed.
• Stir together the ice water and vinegar in a measuring cup, then pour the liquid over the dough as you pulse the machine, forming a moist dough. Scrape the dough onto a work surface, shape into a log and divide into thirds. Press each piece into a disk, wrap tightly and refrigerate overnight.
• The next day, working between sheets of parchment, roll each piece of dough into an 8-inch circle that's a scant 1/4-inch thick. Cut each piece into a 7-inch circle, cover with the parchment and refrigerate for at least 1 hour or up to 2 days. (Leftover dough can be used to make a smaller pie or to make cookies.)
• Make the filling: Stir the sugar, flour and zest together in a large bowl, then stir in the butter and lemon juice. Quarter and core the apples, leaving the peel on, and slice them very thinly; a food processor or mandoline works best here. (You want about 5 loosely packed cups.) Add apples to the large bowl, and use your hands to toss until they're coated with the sugar mixture.
• Assemble the pies: Working with 1 circle of dough at a time, peel away the top sheet of paper, and lightly score a 4-inch circle in the center of the dough. Pile apples inside the circle, pressing to make a compact mound about 1 1/2 inches high and leaving a bare border of about 1 1/2 inches.
• The final pie should have five sides, and all the corners should point in the same direction. For this effect, lift a piece of the dough up, and hold it against the apples. Lift a piece of dough next to it, press it against the apples and grab the triangle of dough that forms where the two layers meet. Holding the point of the triangle, pull it down so that the tip of the triangle is at the base of the pie. With your index finger, firmly press the point of the triangle into the base to seal it in place. Continue until you've got a pentagonal pie.
• Repeat with the remaining circles of dough and apples. (You may have some apples leftover.) Refrigerate the pies while you heat the oven. (You can cover and refrigerate the pies for up to 1 day.)
• Center a rack in the oven, and heat it to 375 degrees. Line a baking sheet with parchment paper, and transfer the pies to the sheet.
• Stir a splash of water into the egg, and brush a little on the sides of the pies, avoiding the top rim of the pie dough. Sprinkle the sides with Demerara sugar.
• Bake the pies until the crust is beautifully golden and a knife inserted into the apples meets little resistance, about 45 minutes. Transfer the sheet to a rack, and allow the pies to cool for at least 30 minutes before serving.