In a large mixing bowl, combine gluten-free all-purpose flour and salt. Cut in the cold, cubed butter until the mixture resembles coarse crumbs. Gradually add the beaten egg and cold water, mixing until a cohesive dough forms. Wrap the dough in plastic wrap and refrigerate for 30 minutes.

In a skillet over medium heat, sauté the chopped onions and bell peppers in a bit of oil until softened, about 5-7 minutes. Add the shredded chicken and cumin, mixing to combine. Stir in the cheese until melted. Remove from heat and let cool.

Preheat the oven to 375°F. Roll out the chilled dough to about ⅛ inch thick. Cut out circles, place filling on one half, fold over, and seal the edges firmly.

Arrange sealed empanadas on a parchment-lined baking sheet. Brush with a beaten egg and bake for 20-25 minutes until golden brown and crispy.