In a large mixing bowl, whisk together the flour and salt. Add the cold, cubed butter and cut it into the flour using a pastry blender, your fingertips, or a food processor until the mixture resembles coarse meal with some pea-sized pieces of butter remaining.

Gradually add the ice water, 1 tablespoon at a time, mixing gently until the dough just comes together. Be careful not to overmix. Divide the dough into two equal portions, flatten each into a disk, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes (or up to 2 days).

In another large bowl, combine the sliced apples, granulated sugar, brown sugar, flour, cinnamon, nutmeg, lemon juice, and vanilla extract. Toss gently to ensure the apples are evenly coated. Set aside while you roll out the dough.

Preheat your oven to 400°F (200°C). Line a 13x18 inch (half-sheet) rimmed baking sheet with parchment paper, leaving an overhang on the two longer sides to help lift the pie out later.

On a lightly floured surface, roll out one disk of dough into a large rectangle, roughly 15x20 inches, or large enough to cover the bottom and extend slightly up the sides of your prepared baking sheet. Carefully transfer the dough to the baking sheet.

Pour the apple filling evenly over the bottom crust, spreading it out to the edges. Dot the top of the apples with the small pieces of cold butter.

Roll out the second disk of dough into another 15x20 inch rectangle. Carefully place it over the apple filling. Trim any excess dough, leaving about a 1/2-inch overhang. Fold the top crust overhang under the bottom crust overhang and crimp the edges together to seal.

Cut several slits or decorative vents in the top crust to allow steam to escape during baking. In a small bowl, whisk together the beaten egg and milk/water to create the egg wash. Brush the top of the pie generously with the egg wash and sprinkle with coarse sugar.

Bake for 20 minutes at 400°F (200°C). Then, reduce the oven temperature to 375°F (190°C) and continue baking for another 40-50 minutes, or until the crust is deeply golden brown and the filling is bubbling thickly. If the crust starts to brown too quickly, you can loosely tent it with aluminum foil.

Once baked, transfer the slab pie to a wire rack and let it cool completely before slicing and serving. This allows the filling to set properly, typically 2-3 hours.