Using a very sharp knife, slice each chicken breast in half horizontally then Put the chicken breasts on a cutting board and lay a piece of plastic wrap over them. Pound the chicken breasts with a meat mallet or rolling pin until they are about a quarter-inch thick.
Put the flour in a shallow platter and season with salt and pepper and mix to combine.
In a separate shallow bowl, beat the eggs with 3 tablespoons of water.
Heat the oil over medium-high heat in a large skillet.
Dredge both sides of the chicken in the seasoned flour, and then dip them in the egg wash to coat completely, letting the excess drip off.
Add the coated chicken to the hot oil and fry for 2 minutes on each side until golden, turning once. Remove to a large platter in a single layer to keep warm.
Add the garlic to the pan and cook for 30 seconds. Add the wine, broth, and lemon juice, simmer for 5 minutes to reduce the sauce slightly.
Mix the melted butter into the flour until combined. Add to the sauce, raise the heat, and bring to a boil to thicken.
Reduce the heat to medium-low and return the chicken to the pan; add the parsley, place the lemon slices on top of the chicken and simmer for 2 minutes to heat the chicken through. Taste for seasoning and Season with salt and pepper to taste before serving.