Preheat your oven to 350°F (175°C).
Open the tubes of refrigerated biscuits and cut each biscuit into quarters.
In a large mixing bowl, combine the coarsely crushed Oreo cookies and the cookies & cream pudding mix. Toss the biscuit pieces in this mixture until they are well coated.
In your baking dish, drizzle half of the melted butter on the bottom. Layer the coated biscuit pieces on top, ensuring they are evenly distributed. Drizzle the remaining melted butter over the top.
Place your dish in the preheated oven and bake for 25 to 30 minutes or until the top is golden brown.
While the monkey bread is baking, combine the softened cream cheese, confectioners' sugar, milk, white chocolate chips, and vanilla extract in a mixing bowl. Whisk until smooth and creamy.
Once the monkey bread is baked, remove it from the oven and allow it to cool for about 5 minutes. Drizzle the creamy white chocolate glaze generously over the warm monkey bread.