Preheat oven to 200 degrees F (95 degrees C). Place a large oven-proof plate on the bottom rack.
Dissolve salt and sugar in water in a 1-quart casserole dish. Place chicken in the marinade. Cover and refrigerate for 30 minutes.
Pat chicken dry and place on a paper towel-lined baking sheet. Let air-dry for 10 minutes.
Place bread in a blender or food processor; blend until crumbly, about 30 seconds. Mix bread crumbs with parsley in a shallow dish.
Place flour in a shallow dish.
Beat eggs with 1 tablespoon olive oil in a separate shallow dish.
Season chicken with salt and pepper. Dredge each cutlet in the flour, shaking off any excess; dip in the egg mixture and coat evenly in bread crumbs.
Heat the remaining oil in a nonstick skillet over medium heat. Add 2 cutlets; cook until golden brown on the outside, no longer pink in the center, and juices run clear, 2 to 3 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Remove warmed plate from the oven; placed cooked cutlets on the plate. Repeat with the remaining chicken.
