Prepare the Chicken: Begin by slicing each chicken breast horizontally to create thin cutlets. Place the chicken between parchment paper and gently pound until even in thickness. Season both sides generously with salt and pepper.
Set Up Breading Station: Arrange three shallow bowls. Add flour to the first bowl. Crack eggs into the second bowl, add Dijon mustard, then whisk until smooth. Place breadcrumbs in the third bowl.
Coat the Chicken: Dredge each piece of chicken in flour, making sure it coats evenly. Dip into the egg mixture, letting excess drip off. Press into breadcrumbs, covering every surface well.
Heat the Oil: Pour vegetable oil into a wide skillet and heat over medium heat. The oil should shimmer gently before adding chicken.
Fry Until Golden: Place chicken carefully into the hot oil. Cook each side until deeply golden and crisp. Transfer to a plate lined with paper towels to drain excess oil.
Finish and Garnish: Sprinkle chopped parsley over the schnitzel and serve immediately with lemon wedges and Dijon mustard.