Preheat your oven to 350°F (175°C). 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, granulated sugar, brown sugar, egg, melted butter, and vanilla extract.
Combine the wet ingredients with the dry ingredients and mix until just combined.
Fill the donut pan about ¾ full with the batter.
Bake for 10-12 minutes or until a toothpick inserted into the donut comes out clean.
Let the donuts cool in the pan for a few minutes before transferring to a wire rack to cool completely.