Make the dough: In a food processor combine the flour and salt. Add the butter and shortening and pulse until it resembles very coarse meal. Add the Cheddar and pulse until it again resembles coarse meal. Transfer the mixture to a bowl. Add the yolk mixture, tossing with a fork until it forms a dough. Add more cold water 1 tablespoon at a time, if necessary. Form the dough into a ball, flatten it into a disc, and dust with flour. Wrap the dough in waxed paper and chill for at least 1 hour or overnight.
Roll out half the dough 1/8-inch thick on a lightly floured surface. Fit it into a 9-inch (1-quart) glass pie plate and trim the edge, leaving a 3/4-inch overhang. Chill the shell and the remaining dough while making the filling.
Make the filling: Preheat the oven to 450 degrees. Peel, core, and cut the apples into eighths. In a large bowl toss together the apples, 3/4 cup of the sugar, the flour, the cinnamon, the nutmeg, the salt, and the lemon juice until the mixture is combined well. Transfer the filling to the shell and dot it with the butter.
Roll out the remaining dough into a 13- by 14-inch round on a lightly floured surface, drape it over the filling, and trim it, leaving a 1-inch overhang. Fold the overhang under the bottom crust, pressing the edge to seal it, and crimp the edge decoratively. Brush the crust lightly with the milk and cut slits in it with a sharp knife to form steam vents. Bake the pie on a large baking sheet in the middle of the oven for 10 minutes. Reduce the temperature to 350 degrees and bake the pie for 20 to 25 minutes more, or until the crust is golden and the apples are tender.
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