A lot of times, Chicken Francaise can be hard to pull off, and have it taste good. This recipe, adapted from the show Food 911 on the Food Network, is a great dish, and it's surprisingly healthy as it doesn't really use all that much oil.
Mix a decent amount of salt and pepper with the flour to make a dredging mixture. You don't need any more than a cup of flour; you're just giving them a light dusting.
Using a meat pounder, pound the chicken breasts flat so that they are about the size of your hand.
Dredge the chicken breasts through the flour. When you pull them out, give them a few good slaps so that the extra flour falls off. You want a very light coating, this isn't supposed to be like batter.
Crack the two eggs into a flat bowl, pierce the yolks with a fork, and scramble them slightly. Combine with about a quarter cup of water to make a light egg wash. It should be pale yellow in color.
Run the chicken through the egg wash. Keep in mind this is a wash, not a bath. You don't want it soaking in egg, just enough to keep it wet and absorb the flour coating.
Heat the quarter cup of olive oil in a large fry pan on medium-high heat, and when it is smoking hot, place the chicken into the pan. Cook for 1-2 minutes on each side. You will know when it is ready to flip when the chicken has a nice, slightly browned color and a crisp texture. Remove the chicken from the pan.
Cut your lemon in half. Using one half, make several paper thin slices of lemon. Save the other half.
Turn the stove up to high heat, and drop the lemon slices into the frying pan. Wait for them to caramellize and take on a slightly gummy texture.
Pour in the 1/2 cup of white wine. I recommend Yellowtail Pinto Grigio because it's not too expensive, you can drink it with the meal, and the sweetness of it counteracts the sour of the lemon. Let the sauce reduce for 30 seconds to one minute.
Sprinkle in salt to taste, and then add in the cup of chicken broth. Allow it to reduce again.
Take your butter, and dredge it in the flour. Break it up into chunks and drop it into the sauce, swirling it around and allowing the sauce to thicken.
Once the sauce has thickened sufficiently, put the chicken breasts back into the pan. Coat them thoroughly with the sauce, and let them cook for about another minute or two to warm them back up.
Transfer to a serving platter, putting a few slices of lemon on top of each chicken breast, and then drizzle the sauce over the whole concoction.
Garnish with flat-leaf parsley, or whatever greenery you think would go good with the meal.