Preheat the oven to 400 degrees F (200 degrees C).

Pound chicken breasts with a meat mallet to an even 1/4-inch to 1/2-inch thickness. Season with salt and pepper on both sides.

Pulse croutons in a food processor until finely crushed. Place crouton crumbs, flour, and beaten eggs into 3 separate shallow dishes.

Dip each pounded chicken breast into flour, shaking off excess. Dip into eggs, letting any excess drip off, then press into crouton crumbs to coat.

Heat olive oil in a large oven-proof skillet over medium heat. Fry chicken in hot oil for 2 minutes. Flip and fry for 2 more minutes. Remove from heat. Top each breast with 2 tablespoons giardiniera and 1 cheese slice.

Cook in the preheated oven until chicken is no longer pink in the center and the juices run clear, 8 to 10 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Top with parsley to serve.