Preheat the oven to 450 degrees. Season the chicken all over with the thyme, 3/4 teaspoon salt and 1/2 teaspoon pepper.
Heat the oil in a large skillet over medium-high heat until very hot, but not smoking. Working in batches, add the chicken to the skillet, skin-side down, and cook, flipping once, until browned on both sides, about 3 minutes per side. Turn the heat down to medium if the skillet starts to get too dark. Transfer the thighs to a baking sheet and roast in the oven until no longer pink near the bone, 10 to 12 minutes more.
While the chicken bakes, add the cabbage, apples, 1/4 teaspoon each salt and pepper to the same skillet over medium heat. Cook, tossing, until slightly wilted, 2 to 3 minutes. Add the broth, vinegar and sugar and continue to cook until the cabbage is tender, the liquid has reduced slightly and the sugar dissolves, 2 to 3 minutes. Remove from the heat and stir in the parsley leaves. Season with additional salt and pepper.
Transfer the cabbage and apples and any juices in the skillet to a serving platter and top with the baked chicken thighs.
Copyright 2016 Television Food Network, G.P. All rights reserved.