Preheat oven to 350°F (175°C). Grease a donut pan lightly with oil or butter.
In a small saucepan, simmer apple cider over medium heat until reduced to 1/2 cup. Let cool slightly.
In a large bowl, whisk together flour, baking soda, baking powder, cinnamon, nutmeg, and salt.
In another bowl, whisk sugars, applesauce, oil, egg, vanilla, and reduced apple cider until well combined.
Add wet ingredients to the dry ingredients and stir until just combined. Do not overmix.
Spoon or pipe batter into the prepared donut pan, filling each well about 3/4 full.
Bake for 12–14 minutes or until donuts spring back when touched and a toothpick comes out clean.
Let cool 5 minutes, then remove from pan and cool on wire rack.
While still warm, brush with melted butter and dip in cinnamon sugar mixture to coat.