The Perfect Apple Pie

For the dough (double recipe if making pie)

1. Cut the butter into 1-inch pieces and freeze for 20 minutes. In a small pitcher, stir together the water and vinegar and refrigerate until needed.
2. In the bowl of a food processor fitted with a metal blade, process the flour, sugar, and salt until blended. Add the butter pieces, pulsing until they are the size of hazelnuts. While pulsing, gradually drizzle in all but 2 tablespoons of the cold water–vinegar mixture through the feed tube until the dough resembles coarse meal.
3. To test it, squeeze a small piece of the dough in the palm of your hand. If it mostly sticks together, you have added enough liquid. If not, gradually pulse in the remaining 2 tablespoons of water-vinegar until the dough holds together when squeezed. Empty the dough into a medium bowl and gently press it into a ball with your hands. For a double recipe, divide the dough into two balls (17 ounces; 480 grams each) and follow the steps below for each of the balls.
4. Wrap the dough with plastic wrap, flatten it into a 6-inch disk, and refrigerate for at least 4 hours. (The dough will keep, double-wrapped with plastic wrap, in the refrigerator for up to 3 days or in the freezer for up to 2 months. Thaw overnight in the refrigerator before using.)
5. Flatten the dough between two large sheets of parchment paper into a rough disk about 9 ½ inches in diameter by pounding it with a rolling pin. Roll out the dough into a round about 13 inches in diameter, occasionally flipping the dough with the parchment, then lifting and smoothing the parchment to avoid creases. If the dough becomes soft and sticky as you roll, transfer the dough and parchment to a baking sheet and freeze for 5 to 10 minutes before continuing.
6. For a 10-inch double-crust pie, peel off the top parchment sheets and trim the edges with a pizza cutter or paring knife to form two neat 13-inch rounds. Cover with the top parchment sheets, stack the covered dough's on a baking sheet, and refrigerate for 30 minutes.
7. Gather the scraps into a small ball, wrap with plastic wrap, and refrigerate for patching the crusts later if needed. The chilled dough rounds are now ready to use in recipes.

For the crust

1. Peel away the parchment from one of the dough rounds, leaving the other round in the refrigerator. Center the dough over a 10-inch fluted pie dish that is 2 1/2 inches deep, easing it across the bottom and up the sides without pressing the edges onto the sides of the dish. Trim the edges of the dough to leave a 1/2-inch overhang. Cover with plastic wrap and refrigerate until needed.

For the Filling

1. Peel, core, and cut both types of apples in half lengthwise (top to bottom). Do not discard the cores and peels. Use a mandoline or sharp knife to slice the apples crosswise into 1/16- to 1/8-inch-thick  slices. Transfer the slices into a bowl large enough to comfortably toss them. Add the sugar, lemon juice, cornstarch, and cinnamon, tossing them gently to avoid breaking up the apple slices too much. Let the apples macerate until they release their juices, about 20 minutes.
2. Meanwhile, coarsely chop the apple cores and peels and transfer them to a medium saucepan. Add the water, cover, and bring to a boil over high heat. Reduce the heat to medium and cook, partially covered, until 1/2 inch of liquid is left in the pan, 10 to 15 minutes. Remove the pan from the heat and strain the mixture through a mesh strainer into a separate medium saucepan, pressing on the solids to extract all of the apple stock.
3. Set mesh strainers over two large bowls, roughly divide the macerated apple slices with their juices between the strainers, and let them drain for 30 minutes.
4. Transfer the apple slices from the strainers to a large bowl, cover with plastic wrap, and refrigerate until needed. Scrape the juices from the two bowls into the apple stock in the pan, and cook over medium- high heat, stirring occasionally, until it is reduced to 1⁄2 cup, 15 to 30 minutes, depending on how much juice the apples have released. Watch closely toward the end to prevent scorching.
5. Remove the pan from the heat, immediately add the butter, salep, and salt, and whisk until blended. (If using glutinous rice flour, cook an additional 2 minutes, whisking constantly.) Scrape the apple sauce into a bowl and refrigerate, uncovered, until it reaches room temperature, 15 to 20 minutes. Once cool, the mixture will be sticky and as thick as porridge.

For the pie

1. Stack about one-quarter of the chilled apple slices in concentric circles over the bottom of the chilled crust. Spread about one-third of the apple sauce evenly over the slices. Continue layering in this manner until you have used all of the apples and sauce, ending with apple slices. The slices will rise about 1 inch above the lip of the dish. If the edges of the dough are very soft, freeze the pie before continuing, uncovered, until the edges are firm, about 10 minutes.
2. Meanwhile, make the glaze. In a small bowl, whisk the egg yolk and heavy cream with a fork until blended.
3. Remove the pie from the freezer and the second dough round from the refrigerator, peel away the parchment, and center the dough over the filling. If needed, trim the dough to make a 1-inch overhang. Tuck the overhang under the bottom crust and press the dough firmly onto the fluted edges of the pie pan to seal it. Brush the dough with the glaze and sprinkle the sugar evenly over the top. Starting about 1 inch from the center, cut 4 evenly spaced slits 2 inches long as steam vents on the top crust. Freeze the pie, uncovered, for 30 minutes.
4. Meanwhile, set a rack in the lower third of the oven, center a rim- med baking sheet lined with aluminum foil on the rack, and preheat the oven to 400°F.
5. Set the pie on the baking sheet in the oven and bake for 20 minutes, then reduce the oven temperature to 350°F and continue baking until the crust is golden brown and the juices are bubbling up through the slits, about 55 minutes longer.
6. Transfer the pie to a wire rack and let cool for at least 3 hours. Serve warm or at room temperature, with salted caramel ice cream if you wish.