Preheat the oven to 350°F (180°C). Prepare two donut trays by spraying them with non-stick spray.
Sift all purpose flour and cocoa powder into a mixing bowl and add brown sugar, baking powder and salt. Mix together until combined.
Add milk, eggs, vanilla extract and melted butter and mix into a batter. Don't overmix.
Add the broken up Oreos and fold them into the batter.
Place the donut batter into a piping bag or ziplock bag and cut off the end with plenty of space for the cookie pieces. Pipe the donut batter evenly into the donut partitions. You can also spoon it in but it does get messy.
Bake for 15 minutes at 350°F (180°C) until lightly browned on top.
Remove from the oven and let the donuts cool in the pans for a few minutes before transferring them to a wire cooling rack to cool completely.
When the donuts are completely cool, add white chocolate to a microwave safe bowl and microwave in 30-second increments bringing it out to stir every 30-seconds until melted.
Dip each donut in the melted chocolate and then place them onto a parchment lined baking sheet.
Sprinkle crushed Oreos on top.
Place the donuts into the fridge for 10 minutes for the topping to set and then serve.