Heat oven to 325° F. Spread the bread in a single layer on 2 rimmed baking sheets. Bake, tossing once, until dry, 20 to 25 minutes; remove and let cool. Increase oven temperature to 425° F.
Meanwhile, melt the butter in a large cast-iron or other ovenproof skillet over medium heat. Add the onion, celery, thyme, ½ teaspoon salt, and ¼ teaspoon pepper. Cook, stirring occasionally, until the vegetables are tender, 15 to 20 minutes. Let cool slightly. Combine the broth and eggs in a medium bowl.
Add the bread and apple to the skillet. Add 3 cups of the broth mixture and toss until the liquid is absorbed. Add up to 1 more cup of the broth mixture to the stuffing as needed. (The stuffing should feel moist but not soggy, and there should be no standing liquid in the bottom of the pan.) Cover with foil. Bake on the top rack until heated through, 25 to 30 minutes. Uncover and bake until golden brown, 10 to 15 minutes more. Serve warm.