Pour 1 cup of apple cider into a small saucepan and simmer over medium heat until it reduces by half, about 10-15 minutes.
In a large bowl, whisk together the all-purpose flour, baking powder, baking soda, ground cinnamon, ground nutmeg, and salt.
In a separate bowl, whisk the melted butter, granulated sugar, eggs, vanilla extract, and the reduced apple cider until smooth and well blended.
Gradually add the dry ingredients into the wet ingredients, folding gently with a spatula just until combined. Be careful not to overmix; a few lumps are okay.
Grease your doughnut pan lightly with butter or a non-stick spray.
Spoon the batter into the doughnut pan cavities, filling each about 3/4 full.
Bake in a preheated oven at 350°F (175°C) for 12-15 minutes, or until a toothpick inserted comes out clean.