01 - Combine flour, salt, and sugar in a large mixing bowl. Cut in cold, cubed butter with a pastry cutter until the mixture forms coarse crumbs with pea-sized pieces of butter. Drizzle ice water over the mixture, one tablespoon at a time, gently tossing until the dough comes together. Divide dough into two discs, wrap in plastic wrap, and refrigerate for at least 1 hour.
02 - In a large bowl, toss peeled and sliced apples with granulated sugar, brown sugar, flour, cornstarch, cinnamon, nutmeg, salt, and lemon juice until apples are evenly coated. Rest for 15 minutes to allow juices to release.
03 - Preheat oven to 425°F (220°C). On a floured surface, roll out one dough disc into a 12-inch circle. Fit the dough into a 9-inch pie pan, leaving a 1-inch overhang. Spoon the apple mixture evenly into the prepared crust, slightly mounding the apples in the center. Dot with butter pieces.
04 - Roll out the second dough disc to a similar size. Position it over the filling. Trim both crusts to a 1/2-inch overhang, tuck under, and crimp edges for a decorative seal. Cut 4 to 5 steam vents in the top crust. Brush with beaten egg and sprinkle with coarse sugar.
05 - Place pie on a rimmed baking sheet. Bake at 425°F (220°C) for 20 minutes, then reduce oven temperature to 350°F (175°C) and continue baking for 35 to 45 minutes, or until the crust is golden brown and the filling bubbles through the vents.
06 - Transfer pie to a wire rack and allow to cool for at least 3 hours for the filling to set. Slice and serve warm or at room temperature with vanilla bean ice cream.