Position two racks near the center of the oven and heat the oven to 425°F. Heat a 10- to 11-inch heavy, ovenproof skillet over medium-high heat. Generously season the chicken on both sides with salt and pepper. Pour 1 Tbs. of the oil into the hot skillet and swirl to coat. Arrange the chicken thighs skin side down in the pan and cook until the skin is golden brown, about 7 min. Turn the chicken over. If a lot of fat has accumulated, carefully spoon it off and discard.

While the chicken browns, toss the beans,onion, carrots, potatoes, and garlic in a large bowl with the remaining 2 Tbs. oil. Season with 3/4 tsp. salt and several grinds of pepper and spread the vegetables on a rimmed baking sheet.

Put the skillet of chicken and the baking sheet with the vegetables in the oven, with the chicken on the higher rack. Roast the chicken until a thermometer inserted in the center of a thigh registers 170°F, 18 to 20 min. Continue to roast the vegetables until very soft and beginning to brown, 8 to 12 min. more.

Meanwhile, remove the skillet from the oven and transfer the chicken to a plate. Spoon off and discard as much fat as possible from the chicken juices, add the wine, set over high heat, and boil until reduced to about 1/4 cup sauce, 4 to 6 min.; it should be syrupy and concentrated in flavor. Swirl in the butter, if using.

Remove the vegetables from the oven and toss them with the basil. Divide the vegetables among four plates. Arrange one or two chicken thighs on the vegetables and drizzle with the pan sauce. Serve immediately.