Preheat oven to 200'
Cut chicken breast in half horizontally. Pound each 1/2 breast to 1/4" thickness using a ziploc bag or plastic wrap to pound out the chicken breast. Salt and pepper chicken and set aside for 15 minutes.
Zest 1 half of the lemon. Cut the lemon in half and juice the zested half. The other half cut into 4 rounds. set aside
Cut 3 tablespoons of butter into 1/2" cubes and toss with 1 teaspoon until butter is coated with flour and no flour remains in the small bowl. Put in fridge until needed.
Put a wire rack in a rimmed baking sheet. Spread the remaining flour in a shallow dish. In another shallow dish whisk eggs with water. Dredge chicken in flour mixture, shaking to remove excess flour. transfer to wire rack.
Heat oils in a 12" skillet over medium-high heat until shimmering. Working with 1 chicken cutlet at a time coat in egg wash and place in skillet. You might need to cook in 2 batches. Reduce heat to medium and cook on 1 side for about 3 minutes until golden brown flip and cook other side for 3 minutes. Repeat with the rest of the chicken.
Discard the oil and wipe out the skillet with paper towels. Return skillet to heat and add lemon slices along with rest of the butter. Cook flipping until lightly browned on both sides about 3 minutes. Transfer to a plate and set aside.
Add garlic to empty skillet, cook stirring constantly until fragrant. Be careful that it doesn't burn. About 30 seconds. Stir in wine and scraping any brown bits from the bottom of the skillet. Increase heat to medium high and bring to a rapid simmer and cook until wine is mostly evaporated. About 3 minutes.
Stir in broth and continue to simmer, stirring occasionally. Until liquid is reduced to 2/3 cup, about 8 minutes.
Reduce heat to low and stir in zest and juice. Whisk in floured butter cubes a couple at a time. Sauce with thicken to the consistency of heavy cream this takes about 2 minutes. Off heat stir in parsley, salt and pepper to taste.
Cut lemon slices in half and scatter over cutlets. Spoon sauce over cutlets and serve!