01 -  Preheat oven to 350°F. Lightly grease two standard donut pans. 
 02 -  In a small saucepan over medium heat, simmer apple cider until reduced to 1/2 cup. Allow to cool slightly. 
 03 -  In a large bowl, whisk together all-purpose flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg. 
 04 -  In a separate bowl, whisk together reduced apple cider, granulated sugar, brown sugar, eggs, unsweetened applesauce, vegetable oil, and vanilla extract until smooth. 
 05 -  Add wet ingredients to dry ingredients and mix until just combined; avoid overmixing. 
 06 -  Spoon or pipe the batter into prepared donut pans, filling each cavity about three-quarters full. 
 07 -  Bake in the preheated oven for 10 to 12 minutes or until a toothpick inserted into a donut emerges clean. 
 08 -  Allow donuts to cool in the pans for 5 minutes before transferring to a wire rack. 
 09 -  In a shallow bowl, mix granulated sugar and ground cinnamon. Brush each warm donut with melted butter, then dip in cinnamon sugar to coat thoroughly.