Make the bread pudding

1. In a large mixing bowl, whisk together the eggnog, milk, sugar, eggs, vanilla, cinnamon, nutmeg, and salt until smooth.
2. Add the cubed bread to the bowl and fold gently so every piece soaks in the custard. Let the mixture rest for 10–15 minutes so the bread absorbs the liquid (longer if you want a more custard-like interior).
3. Transfer the soaked bread into the prepared baking dish, pressing lightly to level the top.
4. Bake uncovered for 35–40 minutes, or until the top is golden and the center is mostly set but still has a slight jiggle. If the surface begins to brown too quickly, loosely cover with foil for the remaining bake time.
5. Remove from oven and let rest a few minutes while you make the sauce.

Make the eggnog sauce

1. In a small saucepan, combine the eggnog, sugar, and cornstarch; whisk until the cornstarch dissolves.
2. Heat the mixture over medium, stirring constantly, until it comes to a simmer and starts to thicken.
3. Meanwhile, beat the extra egg in a small bowl. Slowly ladle about 1/3 of the hot eggnog mixture into the beaten egg while whisking continuously (this tempers the egg and prevents curdling).
4. Pour the tempered egg mixture back into the saucepan and continue to cook, stirring, until the sauce returns to a simmer and reaches a thicker, custard-like consistency.
5. Remove from heat and whisk in the butter until melted. Stir in the spiced rum if using.
6. Keep the sauce warm and serve it spooned over slices of the bread pudding.