Heat a skillet to medium-high heat. Bring a pot of salted water to a boil. Rinse and pat dry the chicken. Dredge first in flour, then in egg, allowing the excess to drip off, and then again in flour. Salt and pepper the chicken breasts.
Heat the olive oil in the skillet until hot but not smoking. Lay the chicken breasts in the pan and allow to sear, two minutes on each side. For optimal browning, do not move the chicken while it is browning.
Turn the heat down to medium low and add the wine to the pan. Allow to cook, making sure that there is always liquid in the bottom of the pan so that the chicken does not burn. Turn once.
Meanwhile, cook the spaghetti in the boiling water until al dente, about six minutes. Drain, reserving about a cup of cooking liquid, and add the spinach to the pasta.
Squeeze the juice of half a lemon over the chicken, and then remove to a separate plate. Add more wine to the pan if necessary, and scrape up all of the bits on the bottom. Whisk together and add to the spaghetti. Add the juice of the other half of the lemon.
Serve the chicken over the spaghetti, with cheese on the side if desired.