Preheat the oven to 400 degrees F. Make sure there is enough room for a tall pie -- you may need to remove a rack.
Remove the dough discs from the refrigerator and set aside, while you peel, core and cut the apples.
In a large skillet, melt the butter. When the butter is sizzling, toss in the apples and stir so they are coated in butter. Cook for about 10 minutes over medium heat, stirring occasionally. If you do not have a pan large enough, you can do this in two batches.
Remove the apples from the skillet (but leave the liquid behind) and put them in a large bowl. Toss the apples in the sugar, cornstarch, cinnamon and nutmeg. Set aside.
Brush the inside of the bottom crust with the beaten egg white. Pile the sauteed apples into the crust, then cover with the remaining dough disc. Trim the crusts, then pinch them together. Using your thumbs and index fingers, crimp the edge into a pretty pattern [or make whatever pattern around the edge you like]. Slice air vents into the top crust. If you like the crust to look shiny, brush with an egg wash; if you like it brown and soft, brush with milk; or you can leave it plain.
Turn the oven down to 375 degrees F. Place the pie pan on a cookie sheet, and bake until the crust is a deep golden brown and the filling is bubbling, about 50-60 minutes.
Let cool completely before serving.