Preheat oven to 425°F (220°C).
In a mixing bowl, combine flour and salt. Cut in butter until mixture resembles coarse crumbs. Stir in ice water, a tablespoon at a time, until dough forms a ball. Divide dough in half and shape into discs. Wrap in plastic wrap and refrigerate for at least 30 minutes.
In another bowl, combine apples, sugar, flour, cinnamon, nutmeg, and lemon juice. Set aside.
On a lightly floured surface, roll out one disc of dough to fit a 9-inch pie dish. Transfer dough to pie dish. Fill with apple mixture and sprinkle shredded cheese on top.
Roll out the second disc of dough and place over the filling. Trim, seal, and flute edges. Cut slits in the top crust to allow steam to escape.
Bake in preheated oven for 45-50 minutes, or until crust is golden brown and filling is bubbly. Cool on a wire rack before serving.