Grease the inside of a 5- to 6-quart slow cooker with the unsalted butter. Add half of the bread cubes, followed by half of the diced apples and half of the chopped walnuts, then repeat the layers with the remaining ingredients.

In a medium bowl, whisk together the eggs, milk, brown sugar, granulated sugar, vanilla extract, cinnamon, nutmeg, and salt until the mixture is smooth and the sugars are mostly dissolved. Pour the custard evenly over the layered bread.

Press the bread down gently with the back of a spoon so the cubes begin absorbing the custard. Cover the slow cooker and let the mixture stand for about 10 minutes before turning on the heat.

Cook covered on LOW for 3 to 4 hours, beginning to check near the lower end of the cooking range because slow cookers vary. The center should be set, a knife inserted near the middle should come out mostly clean, and the center should reach approximately 160°F.

Turn off the slow cooker and let the pudding rest, covered, for 10 to 15 minutes. This short rest helps the custard finish setting while keeping the bread soft and spoonable.

Spoon the warm bread pudding into bowls and top with caramel sauce, whipped cream, or a dusting of powdered sugar. Serve immediately while the apples are tender and the walnut pieces still provide a pleasant contrast.