Preheat your oven to 350°F (175°C). Grease a donut pan.
In a mixing bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In another bowl, combine the apple cider, sugar, melted butter, eggs, and vanilla extract.
Pour the wet ingredients into the dry ingredients and mix until just combined.
Fill the donut pan about ¾ full with the batter.
Bake for 10 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.