Crust: Place flour, sugar and salt in food processor and pulse several times to mix. Remove lid and scatter butter pieces over the dry ingredients. Pulse the machine 4 to 6 times in one-second bursts or until the butter is broken into very small pieces.
Place egg yolk in a 1-cup glass measurer and add just enough of the water to equal 1/4 cup liquid. Using a fork, blend water and yolk. Pour mixture evenly over the flour-butter mixture, making sure it does not clump in one area. Sprinkle grated cheese over this mixture evenly and pulse the machine again, in short bursts, until the pastry starts to form large clumps. Do not over-process. Place a sheet of plastic wrap on the counter and then empty the crumbs into a large mixing bowl.
Using your hands, pack dough and knead it 2 or 3 times. Place dough in center of plastic wrap and flatten into a disk about 3/4". The edges will probably crack slightly; just pinch and mold them back into a smooth disk. Divide dough into two, it in the plastic, and refrigerate until firm enough to roll (about one hour). Remove from refrigerator and proceed to roll out dough. On a floured surface, roll out bottom dough until it's about 1/8" thick. Roll dough onto rolling pin and unroll over your 9" pie dish. Chill for 15 minutes.
Filling: Preheat oven to 425°F. In a bowl, toss sliced apples with lemon juice. Combine sugar, flour and cinnamon, salt and nutmeg; add to apples and toss. Pour into crust and dot with butter.
Roll out remaining pastry to fit top of pie. Place over filling; seal and flute edges and cut slits in top. Beat egg yolk and water; brush over pastry.
Bake at 425°F for 15 minutes. Reduce heat to 350°F; bake 40 - 45 minutes more or until crust is golden and filling is bubbly.
Cool the pie on a cooling rack for 1 or 2 hours before slicing so the juices can firm up.