Apple and Maple Bread Pudding

Name: Apple and Maple Bread Pudding
Ingredients: 6 large eggs; 1/2 cup pure maple syrup (preferably Grade B); 1/2 cup (packed) dark brown sugar; 1 tablespoon vanilla extract; 2 teaspoons ground cinnamon; 1/2 teaspoon fine sea salt; 1 cup whole milk; 1 cup heavy whipping cream; 2 tablespoons apple brandy (such as applejack or Calvados); 1 1-pound loaf pain rustique, all crust trimmed, bread cut into 3/4- to 1-inch cubes (6 1/2 to 7 cups); 3 tablespoons unsalted butter; 2 1/2 pounds Granny Smith apples (about 5), peeled, quartered, cored, cut into 1/3-inch-thick slices (about 7 cups); 1/4 cup pure maple syrup plus additional for brushing (preferably Grade B); 1/4 cup (packed) dark brown sugar