Heat the oil in a cast-iron skillet. Add the chicken, season with salt and pepper, and cook until the meat is golden brown on the outside and no longer pink in the center. Remove chicken from the pan and set aside.
Preheat the oven to 400˚F (200˚C)
To the same skillet, add the onion and garlic. Sauté until translucent. Add the potato cubes and sauté for about 5 minutes. Add the peas and carrots and stir. Add the butter and allow it to melt. Sprinkle the flour on top, covering the vegetables, and quickly stir to avoid lumps. Pour in the chicken broth and bring to a boil to thicken the sauce. Season with salt and pepper and remove from the heat.
Place the pie crust over the chicken and vegetable mixture and carefully seal along the edges of the skillet with your fingers. Brush the pastry with the egg, and cut 3 slits in the top to release steam. Bake for 25-30 minutes, or until golden brown.