Prepare the Dough: In a mixing bowl, combine gluten-free all-purpose flour and salt. Cut in the cold, cubed unsalted butter until the mixture resembles coarse crumbs. Add the egg and gradually incorporate cold water until a soft dough forms. Wrap in plastic wrap and refrigerate for 30 minutes.

Make the Filling: While the dough chills, sauté the chopped onions and diced peppers in a skillet for about 5 minutes until softened. Add the shredded chicken and cumin, stirring to mix. Once heated through, mix in the cheese until melted, then cool slightly.

Assemble the Empanadas: Preheat your oven to 375°F. Roll out the chilled dough to 1/8 inch thick and cut out circles. Place a tablespoon of filling on one half of each circle, fold over, and seal the edges.

Egg Wash and Bake: Place empanadas on a lined baking sheet. Brush with beaten egg for a golden finish. Bake for 20-25 minutes until golden brown. For a crispy texture, fry in hot oil instead of baking.