Preheat the oven to 350° F/175° C. Soak the cranberries in a small bowl of hot water until plump, about 10 minutes, then drain.
Distribute the challah pieces around the buttered baking dish, and then top with the soaked cranberries. In the large bowl, whisk together the beaten eggs, eggnog, rum, and kosher salt. Pour over the challah, thoroughly covering all the bread. Press any dry pieces of bread down into the liquid. Cover with plastic wrap and set aside for one hour.
Stir together the nutmeg and brown sugar, and sprinkle over the challah mixture. Sprinkle the walnuts on top, then bake in the preheated oven until a knife inserted in the middle comes out clean, about 30 minutes. Allow to cool on a wire rack, and serve warm.