Start by flattening the chicken breasts to an even thickness of about 1/2 inch using a meat mallet or rolling pin.
In a shallow bowl, season the flour with salt and pepper. In another shallow bowl, beat the eggs and mix in the grated Parmesan cheese.
Dredge each chicken breast in the flour, shaking off any excess. Then, dip it into the egg mixture, allowing any excess to drip off.
Heat the olive oil and half of the butter in a large skillet over medium heat until shimmering.
Add the chicken breasts to the skillet in batches, cooking for about 4-5 minutes on each side until golden brown and cooked through. Remove the chicken and set it aside on a plate.
In the same skillet, add the chicken broth and lemon juice, scraping up any browned bits from the bottom and bringing the mixture to a simmer.
Add the remaining butter, stirring until melted and combined to create a sauce.
Return the chicken to the skillet, spooning the sauce over the top to coat it well, and allow it to heat through for another 2 minutes.
Serve the chicken hot, garnished with chopped parsley and lemon slices if desired.