Place each chicken breast between sheets of plastic wrap or parchment and gently pound to about 1/4-inch thickness. Cut very large breasts in half horizontally first if needed. Season both sides with salt and pepper.
Place the flour in a shallow dish and season with salt and pepper. In another shallow bowl, whisk the eggs with 2 tablespoons water.
Lightly dredge each chicken cutlet in the seasoned flour, shake off the excess, then dip it into the egg mixture. Let the excess egg drip off.
Heat the olive oil and 2 tablespoons butter in a large skillet over medium-high heat. Cook the chicken in batches for about 2–3 minutes per side, until golden and the thickest part reaches 165°F. Transfer to a plate.
Reduce the heat to medium. Add the white wine to the same skillet and scrape up the browned bits. Simmer for about 2 minutes.
Add the chicken broth and juice of both lemons. Add thin rounds from half of one lemon and simmer for 4–5 minutes, until slightly reduced.
Reduce the heat to low. Whisk in the cold butter pieces one at a time until the sauce is smooth and lightly glossy. Stir in the parsley, salt, and pepper. Return the chicken to the skillet and spoon the sauce over the cutlets. Simmer gently for about 2 minutes.
Transfer the chicken to plates and spoon the lemon butter sauce over the top. Garnish with the cooked lemon slices and additional parsley.