Cook the chicken:
Prepare ingredients: With a sharp knife, cut chicken breasts lengthwise in half. Beat the cutlets to even thickness with a meat tenderizer. Season both sides of chicken with salt and pepper (if you like). In a shallow bowl or flat-bottomed dish, whisk the eggs and a little bit of water together until well blended to make the egg wash. Prepare a separate dish with flour. Line a baking sheet or large casserole with parchment paper.
Bread chicken: Coat the chicken in a thin layer of flour, ensuring to remove any excess. Dip it into the egg batter, allowing any extra batter to drip back into the bowl. Dredge the chicken in flour once more, shaking off any remaining excess. I press down on the chicken with my clean fingers to pat the flour in place all over the surface of the chicken. Be sure to wash your fingers after you touch the chicken and continue cooking. Place each piece of chicken on a plate until all are prepared.
Fry chicken: Heat a large skillet over medium-high heat and add the olive oil. Once the oil is shimmering, hot, but not smoking, carefully place the chicken into the pan. Working in batches, add chicken to the skillet in a single layer and cook until golden brown, about 4-5 minutes. Using tongs, flip and cook, covered, until cooked through, reaching an internal temperature of 165 degrees F, an additional 8 minutes. Reduce the heat if chicken browns too quickly.
Once cooked: transfer the chicken to a plate lined with paper towels and loosely cover it with foil to keep it warm.
Make the sauce:
Blot excess oil: With paper towels, carefully remove any excess oil from the skillet, leaving only a thin layer, along with any bits that may have stuck to the pan. When you blot out the excess oil, make sure you're not touching oil with your fingers, as you could burn your fingers.
Lower heat: Finally, reduce the heat to medium-low. Use a spatula or wooden spoon to scrape and loosen any thing stuck on bottom of the pan. With the pan on medium-low, add butter and garlic. As soon as the garlic has cooked until it is fragrant, add lemon juice and stir.
Reduce sauce: Cook a few minutes while it reduces. Add in the chicken broth, and lemon slices, and stir. Turn the heat back to medium-high. Stir to combine and cook a couple minutes more until it slightly thickens and reduces a little, for about 5 minutes. Keep in mind, this sauce is on the thinner side and does not get very thick. Use a spoon to taste the sauce and if you like, adjust flavor to your liking with a little bit more lemon juice, salt, and some pepper.
Simmer chicken in sauce: Lower heat back to medium-low. Carefully place the cooked chicken in the skillet letting it simmer gently in the sauce, turning it over halfway, about 4 minutes. Spoon sauce to coat the other side and once it's heated through, it's ready to serve.
Serve chicken: Place chicken on a large serving plate or plate it up. Garnish with finely chopped parsley and generously drizzle the sauce over each portion before serving. Enjoy!