Preheat the oven to 350°F (175°C). Grease a Bundt pan with cooking spray.

In a mixing bowl, combine flour, pumpkin puree, granulated sugar, brown sugar, pumpkin pie spice, baking powder, baking soda, and salt.

Add melted butter and beaten egg to the dry ingredients and mix until a dough forms.

Divide the dough into small pieces and roll them into balls.

Layer the dough balls in the prepared Bundt pan, sprinkling with additional pumpkin pie spice and brown sugar as you go.

Bake in the preheated oven for 30 minutes or until golden brown.

While the bread is baking, prepare the cream cheese glaze by mixing softened cream cheese, powdered sugar, milk, and vanilla extract until smooth.

Once the monkey bread is done, remove it from the oven and let it cool for 10 minutes before inverting onto a serving plate.

Drizzle the cream cheese glaze over the warm monkey bread and serve immediately.