Pat the chicken dry with some paper towels. Slice each chicken breast in half, lengthwise. If needed, place the chicken between two pieces of plastic wrap or parchment paper and pound them with a meat hammer to a 1/4-inch thickness.
Using a shallow dish: Mix together the flour, salt, pepper and grated Parmesan cheese. Dredge each piece of chicken in the flour mixture until well coated on each side.
Using a pan or skillet: Heat 1 tablespoon of olive oil and 2 tablespoons butter in a large skillet over medium-high heat. Add half of the chicken pieces, being careful not to crowd the pan. Brown well, for about 3 to 4 minutes on each side. Remove to a plate and cover with foil to keep warm. Wipe out your pan if you need to and repeat the same process with the other chicken breasts until they are all cooked. Do not wipeout the pan after the last batch.
Pour the chicken stock, lemon juice and capers** into the pan and scrape up the browned bits. Reduce the sauce by half. Whisk in the remaining 2 tablespoons of butter. Plate the chicken and serve with the sauce poured over the chicken. Alternatively, you can place the chicken directly back into your pan and baste with the sauce. Garnish with parsley and the lemon slices and serve and enjoy! I like to lightly squeeze some lemon juice over the chicken before serving; this is optional.