Slice chicken breasts in half horizontally, creating four thin portions. If necessary, place between two pieces of plastic wrap and pound to ½-inch thickness.
In a shallow bowl, whisk together milk and egg. In a separate shallow bowl, whisk together flour, cornstarch, salt, and spices.
Dredge each piece of chicken in milk then in the flour mixture. Place on a wire rack and repeat until all chicken is coated. Allow to rest and make the gravy.
When ready to cook, heat 1-inch of cooking oil in a large skillet set over medium heat. Fry the chicken in batches (if necessary) for 6-8 minutes, flipping as needed, until golden brown. Remove to a paper towel-lined plate to drain while cooling.
For the Gravy
In a large saucepan set over medium heat, melt the butter. Whisk in flour and cook for 2-3 minutes, until golden. Whisk in milk and season with salt and pepper, to taste.
Bring mixture to a simmer and cook until slightly thickened, 3 to 4 minutes, then remove from the heat. Taste and season as necessary, then serve with hot chicken.