Begin by placing each chicken breast between two pieces of plastic wrap. Using a meat mallet or rolling pin, gently pound the chicken until it's about half an inch thick.

Season the chicken breasts with salt and pepper on both sides. Spread the flour on a plate and dredge each chicken breast on both sides, shaking off any excess flour.

In a large skillet, heat 2 tablespoons of olive oil over medium-high heat. Once hot, add the chicken breasts and cook for about 3-4 minutes on each side, or until they are golden brown and cooked through. Remove the chicken from the skillet and set aside.

In the same skillet, add 2 more tablespoons of olive oil and 4 tablespoons of butter. Allow the butter to melt before adding the chicken broth, lemon zest, and lemon juice. Stir the mixture to combine, scraping up any browned bits from the bottom of the pan.

Stir in the capers and return the chicken to the skillet. Allow it to simmer in the sauce for about 2-3 minutes, ensuring the chicken absorbs the flavor.

Transfer the chicken to a serving platter, spooning the lemon sauce over the top. Garnish with freshly chopped parsley and serve immediately.