In a large bowl, whisk together flour, salt, and sugar for the crust. Add cold butter cubes and use a pastry cutter (or your fingers) to cut butter into flour until mixture resembles coarse crumbs. Gradually add ice-cold water, 1 tablespoon at a time, mixing until the dough comes together.

Divide dough into two portions, shape into discs, wrap in plastic wrap and refrigerate for at least 1 hour.

Peel, core, and slice the apples thinly. In a large bowl, toss apple slices with granulated sugar, brown sugar, flour, cinnamon, nutmeg, vanilla extract, and lemon juice. Let rest 15–20 minutes to allow flavors to meld.

Preheat oven to 375 °F (190 °C). On a floured surface, roll out one disc of chilled dough to about 12 inches in diameter. Gently press into a 9-inch pie dish, trimming excess dough. Fill crust with the apple mixture, then dot with small pieces of butter.

Roll out the second dough disc and use to cover the pie, sealing and crimping the edges. Cut a few slits in the top crust to allow steam to escape. Brush the top with the egg + milk mixture (egg wash) for golden finish.

Bake the pie for 50–60 minutes, until the crust is golden brown and the filling is bubbling. If edges brown too quickly, cover the rim with foil. Remove from oven and let cool for at least 2 hours before slicing so the filling sets.