Sprinkle a small amount of water on a large sheet of plastic wrap. Place half of the chicken slices on top of the plastic and sprinkle again with water. Cover with another sheet of plastic wrap and pound with a rolling pin or meat pounder until about 1/4-inch thick. Repeat with the remaining chicken.
Beat the eggs with the cheese in a large bowl to blend. Mix the flour with the salt and pepper in a shallow pie plate. Place the chicken in the flour and turn to coat on all sides. Dip into the egg mixture and lift to let the excess drip off. Return to the flour and turn again to coat on all sides.
Heat 2 tablespoons of the olive oil in a large skillet over medium heat. Working in batches, dip the chicken 1 last time in the egg mixture and add to the hot skillet. Cook until golden brown, 2 to 3 minutes per side. Transfer to a large platter, cover loosely with foil, and continue with the remaining scaloppine. Add the remaining tablespoon of oil, if needed.
Heat about 2 inches vegetable oil in a small deep saucepan over medium-high heat until a deep-fat thermometer registers 350 degrees F. Working in batches, add the herbs, patted very dry, a small handful at a time and fry until crisp, about 10 seconds. They will bubble up furiously. Drain on paper towels and sprinkle with salt.
Add the butter, lemon juice, and chopped parsley to the skillet and heat until bubbling. Season with salt and pepper. Spoon a small amount of the sauce over each piece of chicken, top with the fried herbs, and serve at once.
Recipe courtesy of Sara Moulton, "Sara Moulton Cooks at Home", Broadway Books, 2002