Preheat the oven to 350 degrees Fahrenheit. Heat a large skillet over medium-high heat. Season the chicken breasts with salt and pepper and dredge in the flour, shaking off the excess. Add 2 tablespoons oil to the pan. Place the chicken in the skillet smooth-side down and cook until golden, about 5 minutes per side. Transfer to a baking dish and bake until cooked through, about 8 minutes.
Meanwhile, add the onion and apple to the skillet, along with 1 tablespoon oil, if the pan is dry, and increase the heat to high. Cook, tossing, until the onion has wilted slightly and the apple is golden brown, about 2 minutes. Add the vinegar and use a wooden spoon to scrape up the browned bits from the pan. Let the mixture boil until the vinegar becomes syrupy, about 1 minute. Add the chicken broth and return to a boil. Cook until the broth reduces by half.
Remove from the heat, season with salt and pepper and whisk in the butter. Remove the chicken from the oven; add any collected juices to the sauce. Divide the chicken among four plates and spoon the apple, onion and pan sauce over each piece. Sprinkle with the parsley.
Photograph by Antonis Achiellos
Recipe courtesy Food Network Magazine