Whisk the 2½ cups flour and 1 teaspoon salt together in a large bowl. Keep the butter and ice water cold.

Cut the 1 cup cold butter into the flour with a pastry blender until the mixture has pea-size pieces with a few larger flakes. Toss in ice water 1 tablespoon at a time, starting with 6 tablespoons and adding only enough for the dough to hold when pressed. Divide into two discs, wrap and refrigerate for at least 1 hour.

Set an oven rack in the lower third and heat the oven to 425°F (220°C). Peel and core the apples; slice them evenly about ⅛–¼ inch thick. In the second bowl, combine the sugar, 6 tablespoons flour, cinnamon, nutmeg and ¼ teaspoon salt. Toss with the apples and lemon juice.

Lightly dust the work surface with the reserved flour as needed. Roll one dough disc into a 12-inch circle and fit into a 9-inch pie plate without stretching. Layer the apples compactly inside, including their juices, mounding slightly in the center. Dot with the 2 tablespoons butter.

Roll the second disc into an 11–12-inch circle and lay over the filling. Trim the edges to about a 1-inch overhang, tuck under and crimp to seal. Cut 5 small steam vents through the top. Beat the egg with 1 tablespoon water in a small bowl and brush a thin layer over the pastry. Place the pie on a rimmed baking sheet.

Bake for 20 minutes, then reduce the oven to 375°F (190°C) without removing the pie. Continue for 45–60 minutes, until the crust is golden, thick juices bubble through the central vents and a thin knife through a vent meets little resistance from the apples. Shield the edges loosely with foil if browning too quickly. If the filling is not bubbling and tender, continue checking rather than stopping at the clock.

Transfer the pie to a cooling rack with oven mitts. Cool for at least 4 hours so the filling can thicken before cutting into 8 slices.