Step 1

Preparation In the bowl of a food processor, pulse 2 1/2 cups flour, 1 teaspoon salt, and 1 1/2 tablespoons sugar. Add butter and pulse until coarse, pea-sized crumbs appear, about 10 seconds. With the machine running, add vinegar and 1/4 cup ice water and process until the dough just holds together, about 30 seconds. Squeeze a small amount of dough between your fingers and if it is very crumbly, add more ice water, 1 tablespoon at a time (2 tablespoons maximum). Do not over process.

Step 2

Turn dough out onto a lightly floured surface and push together into a rough ball. Knead a few times to combine, then divide into two equal balls. Flatten each ball into a disc with smooth edges (no cracks), cover with plastic wrap, and refrigerate at least 1 hour (or up to overnight).

Step 3

Preheat the oven to 425°F with one rack on the bottom rung and one rack in the center of oven. Place a rimmed baking sheet on the bottom rack to preheat.

Step 4

On a lightly floured surface, roll out one disc of dough into a 13-inch round. Roll the dough loosely around the rolling pin, and then unfurl it into the 9-inch pie pan. Gently lift and settle the dough into the pan. Trim the excess dough using scissors, leaving a 1/2-inch overhang, and transfer to the refrigerator. On a floured piece of parchment, roll out the second disc of dough to the same size as the first and refrigerate both crusts for 30 minutes.

Step 5

In a small bowl, whisk together the egg and milk and set aside. Peel, core, and slice the apples into 1/4-inch thick wedges. Place apples in a large bowl and mix with 1/4 cup flour, 1/3 cup sugar, cinnamon, nutmeg, 1/4 teaspoon salt, and lemon. Fill the dough-lined pan with the apple mixture, packing apple slices as tightly as possible.

Step 6

Brush the rim with egg and milk mixture, reserving the excess. Top with the second dough round and press over apples to minimize excess space between apples and crust. Press the top and bottom crusts together and trim the top crust to a 1-inch edge around pan. Tuck top edge under bottom edge and crimp or decorate as desired. Cut five vents in the top crust for steam. Refrigerate pie for 30 minutes to set the crimp.

Step 7

Brush top crust with egg wash and sprinkle with coarse sugar.

Step 8

Place pie on the preheated rimmed baking sheet and bake until the crust begins to turn golden, 20 to 25 minutes. Rotate sheet, move to center rack, and reduce oven to 350°F. Continue baking until the crust is golden brown and you can see the thickened juices bubbling, 40 to 50 minutes more. Cool on a rack 3 to 4 hours to allow juices to set before slicing.

Step 9

Do Ahead The pie dough can be made up to 3 days in advance and chilled, or frozen for up to 3 months; thaw before using. The baked pie will keep, loosely covered with aluminum foil, at room temperature for up to 2 days.