Butter a 12-cup bundt pan generously with softened butter. Set aside.

In the bowl of a stand mixer, using the paddle attachment, combine 1 cup flour, yeast, salt, and pumpkin pie spice.

In saucepan or microwave safe dish, heat water and butter until melted and warm (120° - 130°).

Add liquids to flour mixture. Blend at low speed until moistened. Add pumpkin, sugar, egg and beat 3 minutes at medium speed.

Switch to the dough hook and mix in the remaining flour a little at a time, to make a soft dough, add more or less flour as needed. Knead the dough for 5 minutes until smooth and elastic.

Turn dough onto lightly floured counter and knead briefly to form smooth, round ball. Coat large bowl with nonstick cooking spray. Place dough in bowl and coat surface of dough with cooking spray. Cover bowl with plastic wrap and allow to rise in a warm, draft free place until dough doubles in size, 60 to 90 minutes.

While dough is rising, mix brown sugar and spices together in bowl. Place melted butter in second bowl. Set aside for the sugar coating.

Remove dough from bowl, and pat into rough 8-inch square. Using bench scraper or knife, cut dough into 64 pieces.

Roll each dough piece into a ball. Working one at a time, dip balls in melted butter, allowing excess butter to drip back into bowl. Roll in brown sugar mixture, then layer balls in bundt pan, staggering seams where dough balls meet as you build layers.

Cover bundt pan tightly with plastic wrap and place in refrigerator overnight. (If you want to bake it immediately, let rise for 30 - 45 minutes in a warm, draft-free place before baking.)

Approximately one hour before you want to bake your Monkey Bread, remove it from the refrigerator and let the dough come to room temperature and the dough will begin to rise. (It will have risen some in the refrigerator overnight.)

Heat oven to 350 degrees. Unwrap pan and bake until top is deep brown and caramel begins to bubble around edges, 30 to 35 minutes. Cool in pan for 5 minutes, then turn out on platter and allow to cool slightly, about 10 minutes.

For the icing: In a mixing bowl, whisk together powdered sugar,milk, vanilla, and maple flavoring until smooth. Add more milk of powdered sugar until the icing is the desired consistency.

Drizzle icing over warm monkey bread, letting it run down the sides of the bread. Serve warm.