Place the chicken breasts between two sheets of plastic wrap and pound them to about 1/4 inch thickness. This ensures quick, even cooking and that delicate, tender bite we want from a classic Francese.
Season both sides with salt and pepper. Place the flour in a shallow dish and set up the eggs in a separate bowl with the optional splash of milk or water to loosen them.
Dip each chicken cutlet into the flour, shaking off the excess. Then dip into the beaten egg, letting any excess egg drip off so you have a light, even coat.
Heat a large skillet over medium heat and add the olive oil. When the oil shimmers, add 2 tablespoons of butter. Once the butter foams, lay in the coated chicken, cooking in batches to avoid crowding.
Sear until the coating is golden and the chicken is just cooked through, about 2 to 3 minutes per side depending on thickness. Transfer the chicken to a plate and keep warm.
In the same skillet, add wine (if using) and simmer for about 1 minute to reduce slightly, scraping up any browned bits from the pan.
Stir in the chicken stock and lemon juice, bringing the mixture to a gentle simmer. Let it bubble for 2 to 3 minutes to meld the flavors.
Return the chicken to the pan and spoon the sauce over the pieces. Let everything simmer together for another 1 to 2 minutes so the chicken finishes cooking through and absorbs some lemony richness.
Remove the pan from heat. Whisk in the remaining 2 tablespoons of butter to create a silky, glossy sauce. If you like, fold in capers for a briny counterpoint.
Taste and adjust with a pinch of salt and pepper. Scatter chopped parsley over the top and squeeze a touch more lemon if you crave extra brightness.
Serve immediately with a side that soaks up sauce nicely—pasta, rice, or buttery mashed potatoes all work beautifully. A final squeeze of lemon over the plate brightens each bite.