Set a wire rack inside a rimmed baking sheet. Cut two 12-inch sheets of plastic wrap.

Cut the chicken breasts in half lengthwise, making 4 roughly equal pieces; they should look like large chicken tenders. Place one sheet of plastic on a clean cutting board. Put a chicken piece in the center of the plastic and top with the other sheet of plastic. Pound the chicken with the flat side of a meat mallet, starting in the center and working your way to the edges, until the chicken is 1/4-inch thick and 6 to 8 inches long. Transfer to the wire rack. Repeat with the remaining chicken, one piece at a time. Liberally sprinkle with salt and pepper. Refrigerate the chicken cutlets uncovered for 1 hour. 

Remove the zest from half of the lemon in wide strips with a vegetable peeler, leaving the bitter white pith behind. Halve and juice the whole lemon.  

Put the flour in a shallow bowl or plate. Dredge the chicken in the flour, shaking off any excess. Heat the oil in a large nonstick skillet over medium-high heat until very hot, about 1 minute. Add 1 tablespoon of the butter and melt. Working in 2 batches, cook the cutlets, adding another tablespoon of butter to the skillet between batches, until golden brown, about 2 minutes per side (it's ok if the chicken is not completely cooked through, it will finish cooking in the sauce). Transfer the cutlets to a plate. 

Reduce the heat to medium. Add the garlic and 1 tablespoon of the butter to the skillet and cook until the garlic is tender and fragrant, about 1 minute. Add the wine and cook until almost completely evaporated, about 4 minutes. Add the capers, lemon zest, 2 tablespoons of the lemon juice, 1/4 cup water and the cutlets with any accumulated juices and bring to a boil. Add the remaining 3 tablespoons butter. Turn the chicken to coat in the sauce and cook until the butter melts and the sauce thickens slightly. Garnish with chopped parsley.