In a large mixing bowl, combine the all-purpose flour, sugar, and salt. Add the cold butter cubes and cut them into the flour using a pastry cutter or your fingertips until the mixture resembles coarse crumbs. A few small butter pieces should remain for a flaky texture.
Gradually add the ice-cold water, one tablespoon at a time. Mix gently until the dough begins to come together. Avoid adding too much water or overworking the dough. Divide it into two portions, shape each into a flat disc, and wrap them separately.
Place both dough portions in the refrigerator for at least 30 minutes. Chilling helps the butter remain firm and makes the pastry easier to roll out.
Wash, peel, core, and slice the apples into evenly sized pieces. Place them in a large bowl and add lemon juice. Toss gently to prevent browning and maintain their fresh flavor.
Add granulated sugar, brown sugar, flour, cinnamon, nutmeg, vanilla extract, and salt to the apples. Mix until the apple slices are evenly coated. Allow the mixture to rest for approximately 10 minutes.
Preheat the oven to 400°F (200°C). Lightly flour a clean work surface and roll out one portion of the chilled dough into a circle large enough to cover a 9-inch pie dish. Carefully transfer the dough into the dish and gently press it against the bottom and sides.
Transfer the prepared apple mixture into the pie crust, arranging the slices evenly. Pile them slightly higher in the center because the apples will shrink as they bake. Cut the tablespoon of butter into small pieces and distribute them over the filling.
Roll out the second portion of dough and carefully place it over the apple filling. Trim the excess pastry, leaving a little overhang. Press the top and bottom crusts together and seal the edges by pinching or crimping them. Cut several small slits in the top crust to allow steam to escape.
Whisk the egg and milk together in a small bowl. Brush the mixture evenly over the top crust. Sprinkle coarse sugar over the surface if desired. This helps create a beautiful golden color and slightly crunchy finish.
Place the pie on a baking sheet to catch any bubbling juices. Bake at 400°F (200°C) for 20 minutes. Reduce the temperature to 350°F (175°C) and continue baking for another 35–45 minutes, or until the crust is golden brown and the filling is bubbling through the vents.
Remove the pie from the oven and place it on a cooling rack. Allow it to cool for at least 2 hours before slicing. This resting time helps the filling thicken and prevents the slices from falling apart.
Cut the pie into even slices and serve slightly warm or at room temperature. Add vanilla ice cream, whipped cream, or caramel sauce for an extra-special homemade dessert.