Prepare the chicken breasts: Cut the boneless, skinless chicken breasts horizontally to create thinner cutlets. Use a sharp knife and cut parallel to the board carefully.
Dredge the chicken in flour: Place flour in a shallow bowl and coat each chicken piece evenly by turning and shaking off excess flour.
Slice the lemon: Cut the lemon into slices, not too thin, and set aside for the sauce.
Cook the chicken: Heat a pan on high heat, add cooking oil, and sear the chicken pieces three at a time. Season lightly with salt (due to salty capers) and cook covered for about 2 minutes per side until golden and cooked through.
Prepare garlic butter and beurre manié: Melt half garlic butter and half beurre manié (butter mixed with flour) in the pan to create a roux-like sauce base. Cook the flour mixture for about 10 minutes while stirring.
Add lemon slices and white wine: Add lemon slices and pour in dry white wine. Let it reduce almost dry while stirring and turning the sauce.
Add chicken broth and reduce: Pour in chicken broth to desired height, increase heat, and let the sauce reduce slowly. Season with pepper and adjust salt carefully.
Finish the sauce: Whisk the sauce to incorporate the beurre manié fully, creating a smooth, glossy sauce. Add capers and chopped parsley.
Reheat Parisian potatoes: Warm the Parisian potatoes in a pan with a little salt, pepper, and parsley.
Plate and serve: Arrange the cooked chicken on plates, spoon the sauce over, add Parisian potatoes on the side, and garnish with fresh parsley. Serve hot.