How to Make It
In a large cast-iron or heavy skillet, heat the oil over medium high heat until hot, then stir in the onion, garlic, carrot, 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook, stirring occasionally, until golden, 8 to 10 minutes. Stir in the flour and cook, until incorporated, about 2 minutes. Pour in the stock and bring to a boil, then stir in the chicken. Season the filling with salt and pepper to taste. Let the filling cool to warm.
Preheat the oven to 375°F. Place the empanada disks on a work surface and divide the filling between the disks. Brush some of the egg around the outer edges of the disks, then fold the disks over the filling to form half-moons. Crimp the edges with a fork.
Brush the tops of the hand pies with some of the egg then cut a steam vent in each. Place the hand pies on a baking sheet and bake until golden, 20 to 25 minutes. Serve warm.