Place 4 medium boneless, skinless chicken breast cutlets on cutting board. Cover chicken cutlets with sheet of plastic wrap, then pound each cutlet with meat mallet or rolling pin until cutlets are approximately ¼-inch-thick at all points.
Add ¼ cup all-purpose flour, salt, and freshly cracked black pepper to shallow bowl. Whisk to incorporate. Set aside.
Crack 2 large eggs directly into medium mixing bowl and add 2 tablespoons water. Whisk ingredients together well until fully incorporated into smooth egg wash. Set aside.
Place large skillet on stovetop over medium-high heat. When skillet is warm, add oil of choice (approximately 2 to 4 tablespoons, as needed). Continue heating skillet until oil is hot and shimmery.
Uncover chicken cutlets and discard plastic wrap. One by one, dredge chicken cutlets through flour mixture. Gently press cutlet into flour to coat completely, then shake any excess flour back into bowl. Repeat on other side.
Dip each floured cutlet into egg wash, coating both sides of cutlet completely. Let any excess drip back into bowl, then place cutlets in skillet.
Pan-fry cutlets 2 minutes or until golden underneath, then flip cutlets over and fry 2 minutes on other side.
When cutlets are golden on both sides, carefully remove cutlets from skillet and transfer to large plate. Set aside. Reduce heat under skillet to medium.
Add 2 tablespoons salted butter and 1 tablespoon all-purpose flour to small mixing bowl. Use fork to mash ingredients together until butter and flour combine into paste. Set aside.
Add 1 tablespoon salted butter to skillet and let melt, then add lemon coins. Sauté lemon coins 1 to 2 minutes, then flip coins over and sauté 1 to 2 minutes more.
Pour in ½ cup dry white wine and stir to incorporate, making sure to scrape up any browned bits of food that may have stuck to bottom of skillet.
After 30 seconds, or once smell of alcohol has lessened, add ½ cup low-sodium chicken broth and 1 tablespoon fresh lemon juice. Stir to incorporate, then bring mixture to simmer.
Add butter-flour paste to skillet and stir to incorporate. Simmer 1 minute or until paste is fully incorporated, stirring constantly.
Retun chicken cutlets to pan. Turn cutlets over 3 to 4 times to coat all sides in sauce, then simmer cutlets in sauce 1 minute.
After 1 minute, transfer cutlets to serving plates. Spoon additional sauce over tops of cutlets, garnish with chopped fresh parsley if desired, and serve warm.