Make the chicken. Butterfly the chicken breasts, then slice them down the center to create 2 cutlets each. Cover the cutlets in plastic wrap and pound each with a mallet or rolling pin until about ⅓-inch thick.
In a small bowl, combine the garlic powder, paprika, cayenne, salt, and pepper. In a shallow bowl, whisk together the flour and 1 tablespoon of the spice mixture. In a separate shallow bowl, lightly beat the eggs.
Set a cooling rack inside a rimmed sheet pan.
Sprinkle the chicken with the remaining spice mixture. Working with one piece at a time, dip the chicken into the egg, turning to coat and allowing the excess to drip off. Then press both sides of the chicken into the flour mixture to adhere. Transfer to the prepared rack to set for at least 15 minutes.
Pour 1 inch vegetable oil into a large cast-iron skillet. Heat over medium-high. Once the oil is glistening, working with two pieces at a time, add the chicken and cook until golden, 3 to 4 minutes per side. Transfer to a paper-towel-lined plate. Repeat with the remaining chicken, adding more oil to the skillet as needed to maintain 1 inch.
Make the gravy. Melt the butter in a small saucepan over medium-low heat. In a small bowl, whisk the flour with ½ cup of the stock until smooth. Once the butter has melted, increase the heat to high and pour the flour mixture. Cook, whisking constantly, until the mixture is simmering. Reduce the heat to medium and gradually whisk in the remaining stock, milk, garlic powder, thyme, salt, and pepper until thick. Adjust seasoning to taste. Keep the gravy warm, whisking occasionally over low heat until ready to serve.
