Prepare the chicken. Place chicken breasts on a cutting board and cover them with plastic wrap. Then, pound them with a meat mallet or rolling pin until they are evenly thick (approximately ½ inch thick). Next, transfer the chicken to a zipper-top bag or shallow dish.
Make the marinade. Whisk together the olive oil, white wine vinegar, salt, pepper, brown sugar, and paprika in a small bowl. Then, pour the marinade over the chicken, ensuring the chicken is completely coated. Seal the bag, cover the dish, and marinate at room temperature for 30-45 minutes.
Preheat the skillet. Heat a cast iron skillet over medium heat. Add 1 tablespoon of olive oil and swirl to coat the pan.
Cook the chicken. Remove the chicken from the marinade, letting the excess drip off. Then, place the chicken in the hot skillet and cook for 3 minutes on each side until it develops a deep golden-brown crust. Reduce the heat to low and cook for 3-4 minutes on each side until the internal temperature reaches 165°F.
Rest and serve. Transfer the cooked chicken to a plate and let it rest for 5 minutes to retain its juices. Serve the chicken whole or slice it for salads, soups, or a main course. Garnish with fresh parsley before serving.