Either cut the chicken breast halves horizontally once or twice, making them really thin, or put each chicken breast between two pieces of plastic wrap and pound them with a meat hammer to 1/4-inch thickness
Mix together the flour, salt, pepper, and grated parmesan
Rinse the chicken pieces in water
Dredge them thoroughly in the flour mixture, until well coated.
Heat 2 tbsp olive oil and 2 tablespoons of the butter in a large skillet on medium high heat
Add half of the chicken pieces, but do not crowd the pan
Brown well on each side, about 3 minutes per side
Remove the chicken from the pan and reserve to a plate
Add 2 tbsp olive oil and 2 tbsp butter to the pan and cook the other breasts in the same manner
Remove from pan
Cover with aluminum foil and keep warm in the oven while you prepare the sauce
Add the chicken stock (or white wine), lemon juice, garlic, and capers to the pan
Use a spatula to 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