Place the chicken breasts on a board covered with baking paper. Place a second sheet of baking paper over the top and bash each chicken breast with a rolling pin to flatten.
In a food processor, pulse the bread, parmesan, parsley, and salt and pepper until crumbs form. Transfer mixture to a bowl; place the lightly beaten egg into a separate bowl.
One at a time, dip each chicken breast into egg, then dip into breadcrumb mixture.
Heat 1 tablespoon oil in a large nonstick pan over medium heat. Place 2 chicken breasts in the pan and cover with a circle of greaseproof paper. Place a cast iron pan on top of the chicken to weigh them down - this helps make the crust extra crispy. Cook the breasts, two at a time until they are cooked through, which should take 2 to 3 minutes per side.