Zest ½ the lemon, and set the zest aside for later. Cut the lemon in half. Squeeze the zested lemon half, and measure out 1 ½ tablespoons of lemon juice. Thinly, slice the remaining lemon half.
One by one, place the chicken breasts on one side of a large sheet of plastic wrap, and fold the other side over top. Pound it with a meat mallet until the chicken breast is a uniform thickness, about ⅓-inch thick.
Mix ½ cup of flour with the salt and pepper on a large plate. Crack the eggs into a wide, shallow bowl, and whisk well with a fork.
Dip the chicken breasts first into the egg, then into the flour mixture to coat.
Heat 1 ½ tablespoons of olive oil in a 12-inch nonstick skillet, then add 2 chicken breasts. Sautée until lightly browned on the bottom, about 5 minutes. Flip and cook for another 4 minutes or so, until the other side is lightly browned and the chicken is cooked through.
Remove the chicken breasts from the pan and set them aside, covering them with foil to keep warm. Add the remaining olive oil to the skillet, then repeat the cooking process with the other 2 chicken breasts.
Whisk ½ tablespoon of flour into the chicken broth until dissolved. Add the white wine to the pan, and let it simmer for 5 seconds. Add the chicken broth, butter, lemon juice, and lemon zest to the pan.
Simmer for 5 minutes, until the sauce thickens slightly. Taste and season it with salt and pepper, if necessary.
Add the chicken breasts back to the pan, along with the lemon slices and half of the parsley, and simmer for 2 minutes. Pour the extra sauce over the chicken, and garnish with the remaining parsley to serve.