Carefully slice the chicken breasts in half horizontally so that there are four pieces of chicken. If they are still fairly thick, you might want to place a piece of plastic wrap over each piece and use a flat meat mallet to pound them roughly 1/4- to 1/2-inch thick so they cook evenly and quickly. Season the chicken breasts on both sides with salt and pepper, about 1/2 to 1 teaspoon of each total.
2 large boneless skinless chicken breasts, 1 teaspoon coarse kosher salt, 1/2 teaspoon freshly ground black pepper
Place the flour on a plate or in a shallow dish and dredge each piece of chicken in it on both sides, shaking off any excess flour. Set aside.
1/3 cup all-purpose flour
In a large skillet, heat 2 tablespoons of the olive oil over medium-high heat. Add the chicken and cook for 3-4 minutes until a nice golden brown crust forms on the bottom side. Flip the chicken over and cook on the second side until cooked through. You may need to work in batches. The chicken is done when it reaches 165°F when tested with an instant digital meat thermometer. Remove the chicken from the pan and set on a plate while you make the sauce.
4 Tablespoons olive oil
Add the remaining oil to the same pan with the drippings from the chicken. Add the garlic and cook for 30-60 seconds until fragrant. Add the chicken broth, capers, and lemon zest, scraping up the browned bits from the bottom of the pan with a wooden spoon. Cook for 5-7 minutes until the liquid has reduced roughly by half.
2 cloves garlic, 1 cup chicken broth, 2 Tablespoons capers, 2 teaspoons lemon zest
Add the lemon juice and cook for another 1-2 minutes until slightly reduced. Add the butter and stir until melted, then return the chicken to the pan with the sauce. Spoon the capers and sauce over the chicken before serving.
1/3 cup fresh lemon juice, 2 Tablespoons salted butter, 2 Tablespoons chopped fresh Italian parsley