Preheat the oven to 350°F. Line a baking sheet with parchment (or foil). Using a meat pounding gadget (or a wine bottle or a rolling pin), gently pound the chicken cutlets to 1/2-inch thickness.
Put the flour, salt, and pepper in a gallon-size zip-lock bag. Shake to mix. Add the chicken, seal the bag, and shake until the chicken is lightly but evenly coated with the flour.
Beat the eggs in a large, shallow bowl. Put the breadcrumbs on a pie plate. Using tongs, dip each chicken cutlet into the beaten eggs, turning to coat evenly, and letting excess egg drip into the bowl. Then dip each cutlet into the breadcrumbs, turning to coat both sides.
Warm the 2 tablespoons of butter and the olive oil in a large skillet over medium heat. When the butter melts, add the chicken. Cook, undisturbed, until golden brown -- about 2 minutes. Then flip the cutlets to brown the other side for 30 seconds to 1 minute. Transfer the chicken to a parchment- (or foil-) lined baking sheet. Drain the fat from the skillet.
Put the chicken in the preheated oven to finish cooking (5 minutes or less; do not overcook!). Meanwhile, make the sauce: Put the chicken stock, wine, lemon juice, in the skillet. Bring to a boil over medium heat. Then reduce the heat, and let simmer until the sauce reduces and begins to thicken slightly -- about 2 minutes. Off heat, stir in the capers and the butter, swirling the skillet until the butter melts -- about 30 seconds. Stir in the parsley.
Divide the chicken among 4 plates. Spoon sauce over the chicken. Enjoy!