Place the chicken breasts between 2 sheets of parchment paper. Using a rolling pin, pound each cutlet to ¼-inch thickness. Season the chicken on both sides with salt and black pepper.
Add the flour to a shallow bowl. In another bowl, whisk together the eggs until smooth. Coat each piece of chicken lightly in the flour, shaking off any excess, then dip it into the egg mixture until fully coated.
Heat half of the butter and olive oil in a large skillet over medium-high heat. Once hot, add half of the chicken and cook for about 2–3 minutes per side, until golden brown and fully cooked through. Transfer the chicken to a serving platter. Carefully wipe out the skillet, then repeat the process with the remaining butter, olive oil, and chicken.
In the same skillet, melt the butter, then add the lemon slices and the garlic cloves. Cook for 1-2 minutes, then add the flour and stir until combined. Gradually add the stock, white wine, lemon juice, and salt. Simmer for 5 minutes on medium-low heat until the sauce thickens and is reduced by half.
Return the chicken to the skillet with the lemon butter sauce. Place the lemon slices and garlic on top of the chicken and simmer for 2 minutes until everything is well coated with the sauce.
Transfer the chicken francese to a plate and coat with the lemon butter sauce. Garnish with fresh chopped parsley.