Preheat the oven to 425°F. Lightly spray three 6-cavity donut pans with nonstick cooking spray.
In a large bowl, whisk the butter, oil, and sugar together until combined. Add the eggs and vanilla and whisk until well combined.
In a separate bowl, add the flours, baking powder, baking soda, salt, and nutmeg. Stir with a whisk to combine.
Stir half the flour mixture into the wet ingredients, followed by the buttermilk, then the remaining flour. Stir just until combined after each addition. The batter will be thick.
Transfer the batter into a disposable piping bag or zip-top bag with a small corner cut off. Pipe the batter into the prepared donut pans, filling each cavity ⅔ to ¾ full.
Bake for 7-9 minutes, or until donuts spring back when pressed lightly with your fingertip. Set the donuts on a wire rack while you make the glaze.
Make The Glaze
In a medium bowl, combine the confectioners’ sugar, milk, and vanilla. Stir until smooth, adding more milk if needed.
While they are still warm, dip the donuts into the glaze and set them on a wire rack placed over parchment paper. Enjoy the donuts warm or allow them to cool. The glaze will set as they cool.