1 1/2 to 2 pounds turnips, trimmed, peeled, and cut into 1-inch cubes
2 tablespoons honey
Preheat the oven to 375°F.
Place the chicken in a medium bowl and toss with 2 tablespoons of the vinegar, 1 teaspoon salt, and 1/2 teaspoon pepper. Heat the olive oil in a large skillet or Dutch oven over medium-high heat. Place the chicken skin-side down in the skillet and sear for 4 to 5 minutes, then flip the chicken and nestle the turnips in around the chicken. Sprinkle the turnips with 1/2 teaspoon salt.
Place the skillet in the oven and roast for 10 minutes, then flip the turnips (they should be browned on the bottoms). Roast until the chicken is a deep golden brown and registers 165°F on an instant-read thermometer and the turnips are tender, another 10 minutes longer.
Transfer the chicken and turnips to a serving platter. Add the remaining 4 tablespoons vinegar to the hot skillet and scrape up any browned bits. Return to medium heat and add the honey. Stir until thick and syrupy, about 6 minutes. Pour over the chicken and turnips and serve immediately.