Whisk the buttermilk and egg in a shallow dish, then combine 1 1/4 cups flour, salt, 1/2 teaspoon pepper and paprika in another dish.
Dip each cutlet in the buttermilk mixture, press it into the seasoned flour and rest it on a rack for 10 minutes.
Heat 1/2 inch vegetable oil in a heavy skillet to 350°F.
Fry the cutlets in batches for 3 to 5 minutes per side, maintaining at least 325°F, until deep golden and the thickest center reaches 165°F.
Drain the chicken on a clean wire rack and carefully pour off the frying oil, leaving the browned bits in the skillet.
Melt the butter over medium heat, whisk in the remaining 1/4 cup flour and cook for 1 minute until it smells toasted.
Gradually whisk in the milk and remaining pepper, then simmer for 4 to 6 minutes until the gravy coats a spoon.
Spoon a modest amount of gravy over the crisp chicken immediately before serving.
