Preheat the oven to 350 degrees. Grease a 9-by-5 loaf pan/tin with butter, and fill with cubed leftover doughnuts. Set aside.
In a mixing bowl, whisk the eggs, rum, cinnamon, sugar, and vanilla until well blended. Add the milk and cream, whisk to blend, and pour over the doughnuts, turning the top pieces so that all of the doughnuts are soaked in the wet mixture.
Bake for 45 to 50 minutes, or until browned on top and firm in the center. Let cool for 10 minutes in the pan.
Icing: Whisk the icing ingredients together in a small bowl until smooth.
Serve the pudding in thick slices, still warm, drizzled with icing.
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