Slice the chicken breasts horizontally into thinner cutlets if needed. Place the cutlets between sheets of plastic wrap and pound them to about 1/4-inch thickness, then season with the salt and black pepper.

Place the flour in one shallow dish. In a second shallow dish, whisk the eggs with the water until evenly combined.

Dredge each cutlet lightly in flour, shaking off the excess, and then coat it in the egg mixture.

Heat the olive oil and 1 tablespoon of butter in a large skillet over medium to medium-high heat.

Cook the chicken in batches, without crowding the skillet, for about 2–3 minutes per side until golden brown and cooked through. Transfer the cooked cutlets to a plate.

Pour off excess fat if necessary while leaving the browned bits in the skillet. Add the white wine, scrape up the fond, and simmer for 1–2 minutes to reduce slightly.

Add the chicken broth and lemon juice. Simmer for about 3–5 minutes, until the sauce is lightly reduced.

Lower the heat and whisk in the remaining 3 tablespoons of butter until the sauce is glossy. Return the chicken to the skillet briefly to warm it and coat it with sauce.

Garnish with chopped parsley and lemon slices or wedges. Serve immediately.