Make the Crust: In a large bowl, whisk together the flour and salt. Add the cold, cubed butter to the flour mixture. Use a pastry blender or your fingers to cut the butter into the flour until the mixture resembles coarse crumbs. Gradually add the ice water, one tablespoon at a time, mixing until the dough just comes together. Be careful not to overmix.

Chill the Dough: Divide the dough in half, flatten each half into a disc, wrap them separately in plastic wrap, and refrigerate for at least 1 hour. This chilling time is crucial for a flaky crust!

Prepare the Apple Filling: In a large bowl, combine the sliced apples, sugar, flour, cinnamon, nutmeg, lemon juice, and the 2 tablespoons of butter. Toss everything together until the apples are evenly coated.

Roll Out the Bottom Crust: On a lightly floured surface, roll out one of the dough discs into a 12-inch circle. Carefully transfer the dough to a 9-inch pie plate. Trim any excess dough and crimp the edges.

Add the Filling: Pour the apple filling into the prepared pie crust, spreading it evenly.

Create the Lattice Top: Roll out the second dough disc into a 12-inch circle. Use a pizza cutter or knife to cut the dough into ½-inch wide strips. Arrange the strips over the filling in a lattice pattern. To do this, lay half the strips across the pie, then fold back every other strip. Place a strip perpendicular to the folded strips, then unfold the folded strips back over the new strip. Repeat with the remaining strips, alternating which strips are folded back each time. Trim the edges of the strips and crimp them to seal with the bottom crust.

Egg Wash and Sprinkle: Brush the lattice top with the beaten egg wash and sprinkle with the remaining 1 tablespoon of sugar. This will give your pie a beautiful golden color and a touch of extra sweetness.

Bake the Pie: Bake in a preheated oven at 400°F (200°C) for 15 minutes. Then, reduce the oven temperature to 375°F (190°C) and continue baking for another 35-40 minutes, or until the crust is golden brown and the filling is bubbly. If the crust starts to brown too quickly, you can cover the edges with aluminum foil.

Cool the Pie: Let the pie cool completely on a wire rack before slicing and serving. This allows the filling to set properly.