Preheat your oven to 375°F (190°C).
In a mixing bowl, combine flour and butter until the mixture resembles coarse crumbs. Gradually add ice water and mix until dough forms. Divide the dough in half, shape into disks, wrap in plastic, and refrigerate for at least 30 minutes.
In another bowl, mix apples, granulated sugar, brown sugar, cinnamon, nutmeg, and lemon juice.
Roll out one disk of dough on a floured surface and fit it into a pie dish. Fill with the apple mixture and sprinkle shredded cheddar cheese on top.
Roll out the second disk of dough and place it over the filling. Trim and crimp the edges to seal. Cut a few slits in the top crust to allow steam to escape.
Bake for 50-60 minutes, or until the crust is golden brown and the filling is bubbly. Let cool before serving.