Preheat your oven to 350°F (175°C) and grease a donut pan.
In a large bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In another bowl, mix the apple cider, brown sugar, applesauce, vegetable oil, vanilla extract, and egg until well combined.
Pour the wet ingredients into the dry ingredients and stir until just combined.
Fill the donut pan cavities about ¾ full with the batter.
Bake for 10-12 minutes, or until a toothpick inserted into the donuts comes out clean.
Let the donuts cool in the pan for a few minutes before transferring to a wire rack to cool completely.