In a large bowl, whisk together the flour, baking powder, salt, cinnamon, and nutmeg. In a separate bowl, whisk together the apple cider, brown sugar, egg, 3 tablespoons melted butter, and vanilla extract.
Pour the wet ingredients into the dry and stir until just combined. The batter will be thick and slightly sticky. Do not overmix.
Transfer the batter to a piping bag or zip-top bag and pipe it into a greased donut pan or silicone donut mold, filling each about 3/4 full. You should get about 6 donuts.
Preheat the air fryer to 350F. Place the donut mold in the basket and air fry for 9-11 minutes until the donuts are puffed and a toothpick comes out clean. Let cool for 2 minutes before removing from the mold.
Brush each warm donut generously with melted butter, then roll in the combined cinnamon sugar mixture until coated on all sides. Serve warm.