Preheat oven to 350°F (175°C).
Grease a donut pan lightly with butter or non-stick spray.
In a mixing bowl, whisk flour, sugar, baking powder, baking soda, and salt.
In another bowl, combine buttermilk, melted butter, egg, and vanilla.
Pour wet ingredients into the dry mixture and stir until just combined.
Fold in the crushed Oreo cookies gently.
Transfer the batter to a piping bag (or zip bag with a corner cut).
Pipe batter into donut molds, filling each about ¾ full.
Bake for 10–12 minutes or until donuts spring back when touched.
Remove from oven and cool in the pan for 5 minutes, then transfer to a rack.
Prepare the chocolate glaze by mixing powdered sugar, cocoa, vanilla, and melted butter.
Add milk gradually until glaze is smooth and thick.
Dip each cooled donut into the glaze.
Sprinkle crushed Oreos immediately so they stick.
Let set for 10 minutes before serving.