01 - Preheat oven to 400°F. Line a 10x15-inch rimmed baking sheet with parchment paper or lightly grease with butter.
02 - Combine flour, sugar, and salt in a large bowl. Cut in cold butter until coarse crumbs form. Gradually add ice water until dough holds together. Divide into two portions, one slightly larger.
03 - Roll the larger dough portion on a floured surface to fit the bottom and sides of the pan. Transfer carefully and press into place.
04 - Toss sliced apples with both sugars, cinnamon, nutmeg, salt, cornstarch, and lemon juice until evenly coated.
05 - Spread apple mixture evenly over the bottom crust. Roll remaining dough and place on top. Trim edges, pinch to seal, and cut small steam vents in the top crust.
06 - Whisk egg with milk and brush over the entire top crust. Sprinkle with coarse sugar if desired.
07 - Bake for 35–40 minutes until crust is deep golden and filling bubbles through the vents.
08 - Let cool for at least 30 minutes to set the filling. Slice into squares and serve warm or at room temperature.