Slice each chicken breast in half horizontally to create two thin cutlets. Place the cutlets between two slices of plastic wrap and pound to a ¼-inch thickness. Season lightly with salt and pepper.
Create a breading station with three medium bowls. Place the flour in one bowl, the beaten eggs in a second bowl, and then combine the bread crumbs, Parmesan, and seasoning in a third bowl.
Working with one cutlet at a time, use a fork to dip it in the flour, and then into the egg, letting any excess drip off into the bowl. Finally, dip the chicken into the bread crumb mixture, turning to coat it completely. Place it on a baking sheet and repeat with the remaining cutlets.
Heat one side of the Blackstone (or other outdoor griddle) to medium-high. Pour the oil in an even layer, then add the chicken cutlets and cook, flipping once, until golden brown and cooked through, 4 to 5 minutes per side.
Notes
Get ahead--You can bread the chicken up to 30 minutes before cooking. Keep it in the fridge on a wire rack set over a baking sheet. This will help to prevent the coating from becoming soggy.