Preheat oven to 400°F. Rub chicken thighs with 1 tablespoon of the oil, 1 1⁄2 teaspoons of the salt, and 3⁄4 teaspoon each of the chopped thyme and pepper. Place chicken on a baking sheet lined with aluminum foil. Bake 15 minutes. (Chicken will not be cooked through.)

Meanwhile, combine fennel, onion, apple, celery, carrots, thyme sprigs, 2 tablespoons of the oil, 1 teaspoon of the salt, 1⁄2 teaspoon of the pepper, and remaining 1 teaspoon chopped thyme in a large bowl; toss to coat. Arrange vegetable mix evenly around chicken. Bake at 400°F until vegetables are almost tender, about 20 minutes.

Toss cornbread with remaining 1 tablespoon oil, 1⁄2 teaspoon salt, and 1⁄4 teaspoon pepper. Arrange around chicken; bake at 400°F until a thermometer inserted in thickest portion of chicken registers 165°F, about 5 minutes. Increase oven temperature to broil; cook until chicken and cornbread are golden brown, 4 to 5 minutes.