Heat the olive oil on the stovetop in a large skillet over medium heat.
Beat the eggs in a shallow bowl. In a separate dish, mix the flour and Parmesan. Dredge the chicken pieces in the eggs and then in the flour-cheese mixture.
Add 2 tablespoons of the butter to the heated olive oil, and when the butter melts, add the floured chicken breasts. Cook until browned, 3 to 4 minutes on each side. Transfer the chicken to a platter and set aside. Cook your chicken in batches if they don't all fit comfortably in the pan.
Add the remaining 4 tablespoons of butter and the garlic to the skillet drippings. Saute the garlic for 30 seconds, being careful not to burn it. Add the chicken broth and capers to the skillet, stirring to mix. Cook the liquid over medium-low heat until reduced by half, 3 to 5 minutes. Add the vinegar and lemon juice and heat through.
Return the chicken to the skillet, spooning some of the sauce over the chicken. Cover the skillet and cook over medium heat until the sauce bubbles and the chicken is cooked through, an additional 8 to 10 minutes. Sprinkle with chopped parsley and serve.
Recipe adapted from Home Cooking with Trisha Yearwood (c) Clarkson Potter 2010