Butterfly the chicken breasts by slicing each piece horizontally with sharp knife; pound under plastic to 1/8-inch thick. Season with salt and pepper. Set up a dredging station: In three separate shallow dishes place the flour seasoned with salt and pepper, eggs beaten with milk and seasoned with salt and pepper, and the breadcrumbs mixed with cheese and seasoned with nutmeg and salt.
Warm oil over medium to medium-high heat. Fry 2 cutlets at a time for 3 minutes on each side, or to deep golden. Keep warm in a low oven on a wire rack-lined baking sheet.