In a large mixing bowl, combine the eggs, pumpkin, half and half, milk, sugars, rum, salt, spices, and vanilla, stirring to blend.
Lightly grease a 2-quart baking dish or a 9" x 13" pan; if you’re going to refrigerate the pudding before baking, be sure to use a dish that can go from the fridge to a hot oven. Place the cubed bread in the dish in an even layer, and pour the liquid mixture over it. Let it rest at room temperature for 30 minutes, or for up to 24 hours in the refrigerator.
When you're ready to bake the pudding, stir it together to redistribute the custard; quite a bit of it will have been absorbed by the bread. Sprinkle with grated nutmeg, if desired; and bake in a preheated 350°F oven till set and beginning to brown, about 40 to 50 minutes. Serve warm with whipped cream or ice cream; garnish with minced crystallized ginger, if desired.