Preheat your oven to 375°F (190°C).
In a mixing bowl, combine the gluten-free flour and cold, cubed butter. Use a pastry cutter or your hands to mix until the mixture resembles coarse crumbs.
Add ice water, a tablespoon at a time, until the dough comes together. Form the dough into a disk, wrap in plastic wrap, and refrigerate for 30 minutes.
In another bowl, toss the apple slices with sugar and cinnamon.
Roll out the dough on a floured surface into a large circle. Transfer to a baking sheet.
Arrange the apple slices in the center of the dough, leaving a border around the edges. Fold the edges over the apples.
Brush the edges of the dough with the beaten egg.
Bake for 40 minutes or until the crust is golden and the apples are tender.
Let cool slightly before serving.