Place chicken breasts side by side on a cutting board and lay a piece of plastic wrap over them. Pound the chicken breasts with a flat meat mallet or the bottom of a heavy saucepan to about 1/4-inch thick.
Put flour in shallow platter and season with fair amount of salt and pepper. Mix with fork to distribute seasoning.
In wide bowl, beat eggs with 3 tablespoons water to make an egg wash.
Place oil in large skillet and heat over medium-high heat until hot.
Dredge chicken in seasoned flour, and pat off excess. (Reserve flour.) Dip chicken in egg wash to coat completely, letting excess drip off. Add chicken to hot oil and fry about 2 minutes on each side, or until golden, turning once.
Remove cutlets to a large platter (in a single layer) and place in oven to keep warm.
Now toss the lemon slices into pan and cook for a minute or two, until fragrant.
Add wine, stock and lemon juice, and simmer 5 minutes to reduce sauce slightly.
Roll the pat of butter in the seasoned flour and add to skillet. This will help thicken sauce. Stir or whisk to emulsify and dissolve flour.
Reduce heat to medium-low and return chicken to pan. Fish out lemon slices with tongs and place on top of chicken.
Simmer the whole thing gently for about 2 minutes to heat chicken through.
Season with salt and pepper and garnish with chopped parsley before serving.