Make the Crust

In a large bowl, whisk together the flour, salt, and sugar. Add the cold, cubed butter and use a pastry cutter or your fingers to work it into the flour until the mixture resembles coarse crumbs. Gradually add ice water, one tablespoon at a time, mixing until the dough just comes together. Divide the dough in half, shape each half into a disk, wrap in plastic wrap, and chill for at least 1 hour.

Prepare the Filling

In a large bowl, combine the sliced apples, granulated sugar, brown sugar, flour, cinnamon, nutmeg, and lemon juice. Toss until the apples are evenly coated.

Assemble the Pie

Preheat your oven to 400°F (200°C). Roll out one dough disk on a floured surface into a 12-inch circle. Fit it into the pie dish, allowing the excess to hang over the edges. Add the apple filling, arranging it evenly. Dot the filling with the small pieces of butter.

Top the Pie

Roll out the second dough disk and lay it over the filling. Trim the excess dough, leaving about 1 inch overhang. Fold the edges under and crimp to seal. Cut slits in the top crust to allow steam to escape. Brush the crust with beaten egg and sprinkle with coarse sugar, if desired.

Bake

Place the pie on a baking sheet to catch any drips. Bake in the preheated oven for 45-50 minutes, or until the crust is golden brown and the filling is bubbling. If the edges brown too quickly, cover them with foil.

Cool and Serve

Let the pie cool on a wire rack for at least 2 hours before slicing. Serve warm or at room temperature with a scoop of vanilla ice cream or whipped cream.