Place the pieces of bread in a greased 9x13-inch baking dish. Let the bread sit out, uncovered, overnight to slightly harden. (You could also toast the bread to harden it and save the hours. To toast the bread, preheat oven to 350°F, place bread in an even layer in a greased 9 x 13 baking dish, and bake the bread pieces for 5 minutes or until bread is slightly crispy.)
When ready to bake the bread pudding, whisk together the eggs, milk, sugars, pumpkin pie spice, vanilla, and pumpkin puree in a medium bowl.
Pour the mixture over the bread. Press down to coat all the bread. Let it soak for 20 minutes and preheat the oven to 350°F.
After soaking, bake for 35-40 minutes, or until a toothpick in the center comes out clean (or the internal temperature is 160°F.)
10 minutes into the baking, begin making the vanilla glaze. The glaze takes about 20 minutes to thicken.