Preheat oven to 350 degrees. Toss bread with 3 tablespoons melted butter; season with salt and pepper. Spread on a rimmed baking sheet; toast, rotating halfway through, until browned, 20 to 25 minutes.
Heat remaining tablespoon butter in a 12-inch skillet over medium heat. Add leeks; cook, stirring frequently, until translucent, about 5 minutes. Add apples; cook, turning occasionally, until tender, about 5 minutes.
In a large bowl, combine bread, leek mixture, rosemary, and broth; season with salt and pepper. Add egg; stir until thoroughly combined. Transfer to a buttered 1 1/2-quart shallow baking dish.
Bake until golden on top and crisp around edges, 40 to 45 minutes.