Halve the chicken breasts and pound them thin. Season the flour with salt and pepper, then dredge the chicken lightly in the flour.
In a bowl, beat the eggs with the grated Parmigiano Reggiano.
Dip the floured chicken into the egg mixture, coating evenly.
Heat neutral oil in a skillet over medium heat and fry the chicken until golden on both sides, about 3–4 minutes per side. Remove and set aside.
Drain most of the oil from the pan, leaving a thin layer behind.
Add the lemon slices and lightly sear them on both sides. Remove and set aside.
Deglaze the pan with white wine and let it reduce by half.
Add the chicken stock and lemon juice, then bring to a simmer.
Stir in the butter rolled in flour and allow the sauce to thicken slightly.
Season with salt and pepper, then stir in the chopped parsley.
Return the chicken to the pan and coat it with the sauce.
Serve with the sauce spooned over the top and garnish with the seared lemon slices and additional parsley.