Preheat the oven to 350 degrees F. Heat a large skillet over medium-high heat. Season the chicken with salt and pepper and coat in flour, shaking off the excess. Add 2 tablespoons vegetable oil to the pan. Place the chicken smooth-side down in the skillet 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 red onion and apple to the skillet, along with 1 tablespoon vegetable oil if the pan is dry. Increase the heat to high and cook, tossing, until the onion has wilted slightly and the apple is golden brown, about 2 minutes. Stir in the vinegar with a wooden spoon, scraping up the browned bits from the pan. Let the mixture boil until the vinegar becomes syrupy, about 1 minute. Add the chicken broth and bring 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 pan sauce. Divide the chicken among plates and spoon the apple, onion and sauce over each piece. Sprinkle with the parsley.