Butterfly the chicken breasts, then slice them down the center to create 2 cutlets each. Cover and pound each 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 press both sides into the flour mixture. Transfer to the cooling rack to set for at least 15 minutes.
Pour 1 inch of vegetable oil into a skillet. Heat over medium-high. Cook chicken pieces for 3-4 minutes per side until golden. Transfer to a paper-towel-lined plate, repeating as necessary.
Melt the butter in a saucepan over medium-low heat. Whisk flour with ½ cup of stock until smooth. Once butter melts, increase heat and pour in the flour mixture. Cook, whisking until simmering. Gradually whisk in remaining stock, milk, garlic powder, thyme, salt, and pepper until thick. Adjust seasoning and keep warm.
Drizzle the chicken with the gravy and serve.