Preheat oven to 375°F (190°C).
In a mixing bowl, combine flour and butter. Mix until crumbly.
Add ice water gradually, mixing until dough forms. Divide dough in half, wrap in plastic, and refrigerate for 30 minutes.
Roll out one half of the dough and fit into a pie dish. Set aside.
In another bowl, combine apples, sugar, flour, cinnamon, and nutmeg. Mix well.
Pour apple mixture into the pie crust. Sprinkle shredded cheddar cheese on top.
Roll out the second half of the dough and place over the filling. Seal and crimp the edges. Cut slits in the top crust to vent.
Bake for 45 minutes or until crust is golden brown. Let cool before serving.