Prep and Tenderize
Filet chicken breasts horizontally into cutlets, cover with plastic wrap, and pound to ¼-inch thickness. Season both sides with salt and pepper. In a shallow bowl, whisk eggs, water, grated Parmesan, and chopped parsley.Dredge and Dip
Spread flour onto a shallow dish. Heat olive oil and 2 tablespoons of butter in a skillet over medium-low heat until shimmering. Pat cutlets dry, dredge both sides in flour, shake off excess, then dip into the egg mixture, allowing any excess to drip off.Sear the Chicken
Place egg-coated cutlets directly into the hot skillet. Sear in batches for 2 to 3 minutes per side until golden brown. Transfer cutlets to a plate and tent loosely with foil.Build the Sauce
Pour white wine and chicken stock into the skillet, scraping up browned bits. Simmer for 4 to 5 minutes until reduced by half. Remove from heat, whisk in cold butter until emulsified, then whisk in fresh lemon juice to taste and return pan to low heat.Finish and Coat
Taste sauce and adjust salt. Return cutlets to the skillet, spooning sauce over top until warmed through and thickened.Plate and Serve
Plate chicken immediately, spoon generous pan sauce over top, and garnish with lemon slices and fresh parsley.