In the bowl of a large stand mixer fitted with the dough hook, stir together the milk, water, yeast, and sugar.
Let the mixture sit for 5–10 minutes to proof the yeast. The mixture should become foamy. If it does not, the yeast may no longer be active or the milk and water may have been too hot. The liquids should be warm to the touch, like baby bath water.
Add the salt, egg, butter, pumpkin puree, cinnamon, nutmeg, and half of the flour. Mix until combined.
With the mixer running, gradually add the remaining flour, a little at a time, until the dough pulls away from the sides of the bowl. The dough should feel slightly sticky but should not stick to your hands. You may not need all of the flour.
Continue kneading the dough in the mixer for 4–5 minutes.
Lightly grease the dough and mixing bowl. Cover and let rise in a warm, draft-free place for 1 hour.
Lightly grease a Bundt pan with nonstick cooking spray, or line a 9-inch cake pan with parchment paper. Set aside.
Prepare the coating by combining the sugar and cinnamon in a small bowl. Melt the butter in a separate bowl.
Punch down the risen dough and remove it from the bowl. Cut the dough into small pieces, about 1 tablespoon each.
Roll each piece of dough in the melted butter, coat it in the cinnamon-sugar mixture, and place it in the prepared pan. Repeat with the remaining dough.
If you have extra butter or cinnamon sugar, pour or sprinkle it over the dough in the pan.
Cover the pan gently with plastic wrap and let the dough rise for another hour in a warm place.
Preheat the oven to 350°F. Place the pan on a baking sheet to catch any drips.
Bake for 25–35 minutes, or until the top begins to brown and the center is thoroughly baked.
Remove from the oven and let cool in the pan for 10 minutes.
While the monkey bread cools, prepare the glaze by mixing all of the glaze ingredients together until smooth and thick.
Turn the warm monkey bread out onto a serving dish and drizzle with the glaze. Serve warm
Notes
*If you are using a bundt pan, your monkey bread will be done around 25 minutes. If you are using a cake pan, think more like 35 minutes.