Season chicken with salt and pepper.
Fill a 12-inch cast-iron skillet halfway with oil, and heat over medium heat until a deep-fry thermometer registers 350˚.
In a shallow bowl, whisk together flour, salt, and pepper. Reserve 3 tablespoons flour mixture for Apple Cider Gravy. In a medium shallow bowl, whisk eggs.
Dredge each steak into the flour mixture, shaking off excess. Dredge in egg, shaking off excess, and dredge again in flour mixture. Let steaks sit in flour mixture for 15 minutes.
Working in batches, fry chicken until golden brown, turning once, for 3 minutes, or until a food thermometer inserted into the thickest portion of the chicken registers 165˚. Let drain on a wire rack for 2 minutes.
Serve with Apple Cider Gravy. Garnish with black pepper, if desired.
Make Apple Cider Gravy: In a 12-inch cast-iron skillet, cook butter over medium heat until bubbling, about 1 minute. Add flour mixture, and cook, whisking constantly, for 2 minutes. Slowly whisk in milk, apple cider, pepper, and salt. Cook, stirring frequently until thickened and bubbly, about 5 to 7 minutes. Serve immediately
To serve place one chicken breast on a plate and ladle gravy over the top.