In a large bowl, combine the flour and salt. Add the chilled butter and cut it into the flour using a pastry cutter or fork until the mixture resembles coarse crumbs.
Beat the egg with the cold water, then add it to the flour mixture. Stir until the dough begins to come together. Knead briefly to form a smooth dough, then wrap in plastic wrap and refrigerate for 30 minutes.
Make the Filling
Heat olive oil in a skillet over medium heat. Add the chopped onion and garlic, cooking until softened, about 5 minutes.
Stir in the shredded chicken, tomato sauce, olives, cumin, paprika, salt, and pepper. Cook for an additional 5 minutes until the mixture is well combined and heated through. Set aside to cool slightly.
Assemble the Empanadas
Preheat the oven to 375°F (190°C). Line a baking sheet with parchment paper.
Roll out the chilled dough on a lightly floured surface to about ⅛-inch thickness. Cut out circles using a 4- or 5-inch round cutter.
Place a spoonful of the chicken filling in the center of each dough circle. Fold the dough over to form a half-moon shape, and press the edges together to seal. Use a fork to crimp the edges.
Apply the Egg Wash
Brush the tops of the empanadas with the beaten egg for a golden finish.
Bake the Empanadas
Arrange the empanadas on the prepared baking sheet and bake in the preheated oven for 20-25 minutes, or until golden brown.
Cool and Serve
Allow the empanadas to cool slightly before serving.