Place the chicken breasts between two sheets of plastic wrap and pound them to about ½ inch thickness with a meat mallet or rolling pin.
In a shallow dish, add the flour and season with salt and pepper. In another shallow dish, whisk together the eggs and Parmesan cheese until well combined.
Dredge each chicken breast in the seasoned flour, shaking off the excess, then dip it into the egg mixture, ensuring it is fully coated.
Heat the olive oil and 2 tablespoons of butter in a large skillet over medium heat, then add the chicken breasts and cook for about 4-5 minutes on each side until golden brown and cooked through.
Remove the chicken from the skillet and set it aside on a plate lined with paper towels to absorb excess oil.
In the same skillet, add chicken broth, white wine, and lemon juice, scraping the bottom of the skillet to release browned bits. Bring to a simmer and cook for about 3-4 minutes until slightly reduced.
Add the remaining 2 tablespoons of butter to the sauce and stir until melted and well combined. Return the chicken to the skillet and spoon sauce over each piece, allowing everything to heat through for 1-2 minutes.
Transfer the chicken to a serving platter, drizzle with the lemon sauce, and sprinkle with fresh parsley. Serve immediately.