Slice chicken breasts in half crosswise into 3 cutlets and lightly pound each piece between sheets of plastic wrap until an even thickness (about ½" thick or less). Season both sides lightly with salt and pepper. Place flour in a shallow bowl. Working one at a time, place cutlets in bowl and toss to coat in flour. Knock off excess flour and transfer to a plate.

Heat 2 tablespoons olive oil in a large skillet over medium-high. Working in batches if needed to avoid overcrowding, pan-fry the chicken cutlets, without moving them, until nice and golden-brown underneath, about 2 minutes. Turn cutlets over to cook on the other side just until chicken is nearly cooked through, about 30-60 seconds. Remove from pan and transfer to a clean plate.

Add the sliced garlic and remaining 1 Tbsp. olive oil to the skillet and cook, stirring often and reducing the heat if needed to keep garlic from burning, cook until golden brown, about 1-2 minutes. Add wine, chicken stock, and capers and cook, swirling pan and scraping up any browned bits stuck to the bottom of skillet (we call this deglazing) about 2-3 minutes. Reduce the liquid by ½ and then add ½ cup water, followed by the butter. Swirl pan vigorously while butter melts to help it form an emulsion with water, about 1 minute.

Return chicken to skillet and simmer until chicken is cooked through and sauce is thick enough to coat a spoon, about 2 minutes. Remove from heat and stir lemon juice, lemon zest, and ½ of the chopped parsley into sauce. Transfer chicken and sauce to a platter and top with parsley. Serve with lemon wedges.