In a large bowl, combine flour and salt. Cut in butter until mixture resembles coarse crumbs. Stir in cheese. Combine water and vinegar, and gradually stir in until mixture forms a ball. Divide dough in half and shape into balls. Wrap in plastic and refrigerate for 4 hours or overnight.
Preheat oven to 450 degrees F (230 degrees C.) Roll one ball out to fit a 9 inch pie plate. Place bottom crust in pie plate. Roll out top crust and set aside.
In a large bowl, toss apples in lemon juice to prevent discoloration. Drain and discard any excess juice. Stir in sugar and cinnamon. Arrange rows of overlapping apple slices, working from outer rim in. Dot with butter. Cover with top pie crust. Seal and crimp edges with fork, then trim excess dough. Cut a few slashes in top crust to allow steam to escape.
Bake on cookie sheet in preheated oven for 15 minutes. Reduce heat to 350 degrees F (175 degrees C), and continue baking for about 30 minutes, or until golden brown. Remove from oven, brush lightly with beaten egg, and sprinkle liberally with sugar. Bake 5 to 10 minutes more until sugar forms a crisp glaze.