Serve with crusty bread to sop up the juices.
Preheat oven to 450 degrees with rack in upper third. In a large straight-sided ovenproof skillet, heat 1/4 cup oil over medium-high. Add eggplant and cook, stirring frequently, until just beginning to brown, about 4 minutes. Add zucchini, garlic, and remaining 1 tablespoon oil. Cook, stirring, until garlic is fragrant, about 1 minute. Stir in tomatoes and thyme; season with salt and pepper.
Season chicken on both sides with salt and pepper and rub with oil. Add to skillet, skin-side up, and roast in oven until chicken is cooked through, 25 to 30 minutes. Turn oven to broil and continue to cook until skin is golden brown and juices are bubbly, 1 to 2 minutes more. Let rest 10 minutes before serving.