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.