1. Preheat oven to 350ºF. Spread bread cubes in a single layer on a large baking sheet and bake until pale golden and dry, stirring occasionally, about 20 minutes.
2. Grease a 9-by-13-inch baking dish. On medium-high heat, warm oil in a large skillet and cook sausage, breaking up meat and cooking until sausage starts to brown, 6 to 7 minutes. Add onion and celery; continue to sauté until vegetables are soft and sausage is cooked through, about 5 minutes. Stir in herbs and raisins. Add broth, scraping up browned bits on bottom of skillet. Transfer to a large mixing bowl and stir in apples and bread. Season with salt and pepper. Stir in egg. Spread in prepared baking dish. Drizzle with melted butter.
3. Bake, uncovered, until golden brown and crispy on top, about 50 minutes.