Begin by preparing the chicken breasts. Place each chicken breast between two pieces of plastic wrap and gently pound them to an even thickness, about ½ inch thick.
In a shallow dish, combine the flour, salt, and pepper. In another shallow dish, whisk together the eggs and water until well combined.
Dredge each chicken breast in the flour mixture, shaking off any excess. Then dip it in the egg mixture, allowing any excess to drip off. Return the chicken to the flour and coat it again, pressing gently to adhere the flour.
In a large skillet, heat the olive oil over medium-high heat. Once hot, add the chicken breasts in a single layer. 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.
In the same skillet, add the chicken broth, lemon juice, and capers. Scrape any brown bits from the bottom of the pan to incorporate into the sauce. Bring the mixture to a simmer and cook for 2-3 minutes until slightly reduced.
Return the chicken to the skillet, spooning the sauce over to coat. Cook for an additional 1-2 minutes to warm the chicken through.
Serve the chicken piccata on plates, drizzling the lemon sauce over the top, and garnishing with fresh parsley and lemon slices, if desired.