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. Dip the chicken into the egg, then the flour mixture. Transfer to the prepared rack to set for at least 15 minutes.

Pour 1 inch of vegetable oil into a large cast-iron skillet. Heat over medium-high. Once hot, add the chicken and cook until golden, 3 to 4 minutes per side. Transfer to a paper-towel-lined plate. Repeat with remaining chicken.

Melt the butter in a small saucepan over medium-low heat. Whisk the flour with ½ cup of the stock until smooth. Once melted, increase heat to high and pour in the flour mixture. Cook until simmering.

Reduce heat to medium and gradually whisk in remaining stock, milk, garlic powder, thyme, salt, and pepper until thick. Adjust seasoning to taste. Keep warm over low heat until ready to serve.

Drizzle the chicken with the gravy and serve.