On a clean cutting board, spread out 1 sheet of wax paper or plastic wrap. Place 1 chicken breast on the wax paper and top with a second sheet.

Using a meat mallet or rolling pin, pound the chicken breast until it is evenly flattened to about 1/8 inch thick. Repeat with the remaining chicken breast halves. (You may need new sheets of wax paper or plastic wrap for each breast.) Set aside.

In a shallow bowl, combine the flour, 1 1/2 teaspoons of the salt, and the pepper, and stir to combine.

Quickly dredge the pounded chicken in the flour mixture, shaking to remove any excess flour.

Place 1 1/2 tablespoons of the oil in a large skillet over medium-high heat. When the oil gets very hot, but not smoking, add 1 tablespoon of the butter and, working quickly, add half of the chicken and cook for 1 minute, until golden.

Using tongs, turn the chicken and cook until golden and cooked through, about 1 1/2 minutes. Remove the chicken from the skillet and cover to keep warm.

Repeat with the remaining 1 1/2 tablespoons of oil, 1 tablespoon of butter, and pounded chicken, cooking as above. Remove the chicken from the skillet and cover to keep warm.

Add the lemon juice to the skillet, scraping with a wooden spoon to remove any browned bits from the bottom of the pan. When the juice has reduced by half, about 1 minute, add the chicken broth, garlic, and capers, and cook for 5 minutes, until slightly thickened. Add the remaining 1/4 teaspoon of salt, remaining 4 tablespoons of butter, and parsley. Cook, stirring, until heated through and the sauce has thickened, about 1 minute. Do not allow sauce to boil.

Remove from the heat and pour the sauce over the reserved chicken. Serve immediately.