PREHEAT oven to 350°F. Light coat two 3-quart baking dishes with butter or oil.
MELT butter in a large, heavy skillet over medium-high heat. Add sausage, onion, celery, and cook, stirring frequently, until vegetables are soft, about 5 minutes. Add apples, sage and thyme and continue cooking for 5 minutes. Remove from heat and let cool slightly.
WHISK eggs in a large mixing bowl. Stir in contents of skillet, stuffing mix, and chicken broth. Crumble corn bread into bowl and stir in pepper and salt until well-combined. Transfer mixture to prepared baking dishes. Do ahead: The dressing can be prepared up 24 hours in advance, just cover and refrigerate. Bring to room temperature before baking.
BAKE dressing uncovered for 75 minutes, until top is golden-brown and the center is set.