Place a piece of plastic wrap over the chicken, and use a meat mallet to firmly press down on the breast until you hear the breastbone crack. You can pound on the rest of the bird to be sure that everything is as flat as you can get it.
Heat an oven-safe, non-stick pan (or well-seasoned cast iron pan) over medium heat. Add the vegetable oil. Once the pan is hot, place the chicken breast-side-down and weigh it down with a heavy object like a foil-wrapped brick or a dutch oven.
Remove the chicken from the pan with tongs, taking care not to tear the skin. Return the pan to the oven and continue to cook the potatoes for another 10 minutes. Let the chicken rest while the potatoes finish cooking.