For the donuts- Preheat your oven to 325 degrees and spray 2 donut pans with nonstick cooking spray.
In a large bowl, whisk together the flour, sugar, baking powder, baking soda, salt and nutmeg.
Add teh egg, eggnog and applesauce and beat until well combined. The batter will be pretty thick.
Spoon the batter into a piping bag or large resealable plastic baggie. Cut the bottom (or corner), and pip the batter into the wells of the donut pans, filling each about 1/2 full. Bake for 10-12 minutes, or until the donuts spring back when lightly touched.
Transfer the pan to a wire rack and let cool for a few minutes before removing the donuts to cool completely.
For the glaze- While the donuts are cooling, prepare your glaze. In a small bowl big enough to dip the donuts into, whisk together the powdered sugar and the eggnog. If the glaze is too thick, try adding a splash of milk as the eggnog is thicker than milk. Whisk until desired consistency is reached.
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