Heat a large oven safe skillet over medium-high heat. Add 2 tablespoons olive oil. When the oil shimmers, add the chicken, skin side down, and sear until golden, about 5 minutes. Flip and cook 5 minutes more. Remove the chicken from the skillet. Add 2 tablespoons butter, the potatoes, carrots, and a pinch each of salt and pepper. Cook, stirring often, until slightly softened, about 5 minutes. Pour in the wine, deglazing the pan and scraping up any browned bits off the bottom. Simmer the wine for 5 minutes or until reduced slightly. Remove from the heat and nestle the chicken into the potatoes. Add the lemon juice.