If you haven’t already, prepare the pastry and refrigerate for at least one hour, until firm enough to roll.
On a sheet of lightly floured wax paper, roll the larger portion of pastry into a 13 1/2-inch circle with a floured rolling pin. Invert the pastry over a 9-inch deep-dish pie pan. Center it, then peel off the paper. Gently tuck the pastry into the pan, without stretching it, and let the overhang drape over the edge of the pan. Refrigerate. Preheat the oven to 400°F.
While the pie shell chills, make the filling. Combine the apples, brown sugar, lemon juice, cinnamon and nutmeg in a large mixing bowl; toss well to mix. Set aside while you roll the top pastry.
On another sheet of lightly floured wax paper, roll the other half of the pastry into an 11 1/2-inch circle. Turn the filling into the refrigerated pie shell, smoothing the apples with your hands. Dot the top of the pie with butter, dropping the pieces here and there over the apples.
Lightly moisten the rim of the pie shell with a wet finger or pastry brush. Invert the top pastry over the filling, center it, then peel off the paper. Press the top and bottom pastries together along the dampened edge. Trim the pastry with scissors or a paring knife, leaving a 1/2-inch overhang all around, then sculpt the overhang into an upstanding ridge. Make several 2-inch-long slits in the top pastry, at the 12, 3, 6, and 9 o’clock positions; the bottom of each slit should just reach the ege of the pie. Lightly brush the top pastry with milk and sprinkle with granulated sugar.
Place the pie directly on the center oven rack and bake for 30 minutes. Remove the pie from the oven and place it on a large, dark baking sheet covered with aluminum foil. Reduce the oven temperature to 375°F Put the pie on the baking sheet back in the oven and bake for an additional 30 minutes; when the pie is done, you should be able to see the juices bubbling up onto the crust.
Transfer the pie to a cooling rack and let cool for at least an hour before slicing.