Line a large baking sheet with parchment paper and set aside. Measure the cereal into a large mixing bowl.
In a microwave-safe bowl, combine white chocolate chips and butter. Microwave in 30-second intervals, stirring between each, until completely smooth and melted, about 90 seconds total.
Stir vanilla extract into the melted chocolate mixture until combined. Pour the chocolate over the cereal and gently fold with a silicone spatula until every piece is evenly coated.
Spread the coated cereal onto the prepared baking sheet in a single layer. Let cool at room temperature for 5 minutes until the chocolate is no longer warm to the touch.
Transfer the cooled cereal to a gallon-sized zip-top bag. Add the powdered sugar, seal the bag, and shake vigorously for 30-45 seconds until every piece is covered in a white coating.
Add the crushed Oreos to the bag and shake gently for 10 seconds to distribute the cookie pieces throughout the mixture.
Spread the finished puppy chow on the baking sheet to let any excess powdered sugar settle, about 5 minutes, before serving or storing.
