Use a sharp knife to slice the chicken breasts horizontally, butterflying them open. If the breast pieces you are working with are large, cut them each into two pieces. If the pieces are thick, cover with plastic wrap or place chicken in a Ziploc bag and pound the breasts with a rolling pin or meat mallet to an even 1/4-inch thickness.
Season the chicken breasts with Kosher salt and black pepper. Place the flour on a large plate or shallow dish. Place the beaten eggs in a shallow dish. On another large plate, mix together the panko and lemon zest.
Take a chicken breast and dredge it in the flour, making sure both sides are completely coated. Shake off excess flour. Dip the chicken in the egg wash and then dredge into the panko mixture, covering both sides thoroughly. Set aside and repeat process with remaining chicken breasts.
Heat the butter and olive oil over medium-high heat in a large cast iron skillet. Add the chicken to the pan and brown well on each side, about 3 to 5 minutes per side. Remove the chicken from the pan and set aside on a plate.
In the same skillet, make the sauce. Add the garlic to the pan and cook for 1 minute. Stir in the broth; simmer until the liquid is reduced by half, about 5 minutes. Reduce heat to low, then stir in the butter, lemon juice, and capers. Stir until the butter is melted. Season with salt and pepper.
Add the chicken back into the pan and simmer for 3 to 5 minutes. Top with lemon slices and fresh parsley. Serve warm.
