Combine flour, sugar, salt, butter and shortening in a food processor or mixing bowl. Pulse the machine on and off (or cut the ingredients together with two knives) until the mixture is crumbly. Mix in the grated cheese. Add the water and pulse (or toss with a fork) until the mixture begins to clump together. Gather it into a ball, sprinkling with a few more drops of cold water, if needed. Divide dough into two pieces, one slightly bigger than the other, then flatten slightly into two disks, wrap in plastic and chill for at least 30 minutes before rolling out.
Roll out the bigger piece of the dough on a piece of wax or parchment paper sprinkled with a little flour, to a thickness of about 1/8 inch. Fold in half and place in the pie pan with the fold in the center. Gently unfold and fit the dough, without stretching, over the surface of the pan. Trim the edges of the dough, leaving an overhang of about 1/2 inch. Roll out the top crust, which can be slightly thicker than the bottom and set aside.
Preheat the oven to 375°F. Place the prepared apples into a large mixing bowl; toss with the lemon juice. In a small bowl, stir together the sugar, cornstarch, cinnamon and nutmeg until free of lumps. Toss with the apples and pile into the crust, mounding them in the center. Gently drape the top round of pastry over the filling, leaving about 1/2 inch. Press both layers of dough together against the rim of the pie pan, sealing the edges of the top and bottom crusts. Trim the excess dough and press the edges with a fork all around the edge of the pie. Place the pie in the oven (with the rack in the lower third) and bake until crust is golden brown, about 45 minutes. Remove from oven and let cool about 15 minutes before cutting.