Place each chicken breast on a cutting board and slice horizontally into two or three filet (if you cut it into two filet, use a mallet to pound it a bit thinner); continue with each breast. Season chicken with salt and pepper and dredge chicken in flour; shake off the excess.
In a large skillet (I like to use a cast iron pan) over medium-high heat, melt 2 tablespoons each of butter and olive oil. When butter and oil start to sizzle, add 3-4 pieces of chicken and cook for 3 minutes. When chicken is browned, flip and cook other side for 3 minutes. Remove and transfer to plate. Continue with the rest of the chicken (adding equal parts butter and oil as needed) and, when all of the chicken is cooked, remove the chicken to a platter but keep the skillet over medium heat.
Add minced garlic to the pan and stir; then add the lemon juice, stock, wine and capers. Return to stove and bring to boil, scraping up brown bits from the pan for extra flavor. Check for seasoning. Return all the chicken to the pan and simmer for 5 minutes. Remove chicken to platter. Add remaining 2 tablespoons butter to sauce and whisk vigorously. Pour sauce over chicken and garnish with parsley.
Serve warm with juices and accompany with your choice of sides (veggies like green beans, zucchini, broccoli or cauliflower and a starch like mashed potatoes or a simple, light pasta).
