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

Cut the chicken breasts in half lengthwise. Place one sheet of plastic on a cutting board. Put a chicken piece in the center and top with the other sheet. Pound the chicken until 1/4-inch thick. Transfer to the wire rack. Repeat with remaining chicken and sprinkle with salt and pepper. Refrigerate uncovered for 1 hour.

Remove the zest from half of the lemon in wide strips. Halve and juice the whole lemon.

Put flour in a shallow bowl or plate. Dredge the chicken in the flour, shaking off any excess. Heat oil in a large skillet over medium-high heat until hot. Add 1 tablespoon of butter and melt. Cook the cutlets in batches, adding butter between batches, until golden brown, about 2 minutes per side. Transfer cutlets to a plate.

Reduce heat to medium. Add garlic and 1 tablespoon of butter to the skillet. Cook until garlic is tender, about 1 minute. Add wine and cook until almost evaporated, about 4 minutes. Add capers, lemon zest, 2 tablespoons of lemon juice, 1/4 cup water, and cutlets. Bring to a boil and add remaining butter. Turn chicken to coat and cook until sauce thickens. Garnish with parsley.