In a small saucepan, bring apple cider to a gentle boil and simmer over medium heat until thick and reduced to about 1/3 cup, 15-25 minutes. Set aside to cool.
Preheat oven to 350°F. Grease donut pans with butter or nonstick spray.
In a large bowl, mix the flour, baking soda, baking powder, spices, and salt. Set aside.
In a small bowl, whisk the egg, melted butter, granulated sugar, brown sugar, reduced apple cider, milk, and vanilla extract.
Make a well in the center of the dry ingredients. Add the liquid ingredients to the well and fold using a rubber spatula until combined. Batter will be thick.
Spoon or use a bag to pipe the batter into the greased donut pan, filling about halfway full.
Bake for 9-11 minutes until the top is lightly browned and a toothpick into the center comes out clean. Cool for 5 minutes, then transfer to a wire rack to cool completely.
Combine granulated sugar, cinnamon, and nutmeg in a medium bowl. Dunk cooled donuts in melted butter and coat with sugar topping.