Combine apple cider and grated apple in a small saucepan over medium heat and cook, stirring occasionally, until mixture is reduced to about 1/2 cup (roughly 25 minutes)
Allow mixture to cool completely
Stir applesauce, vanilla and apple cider vinegar into mixture, set aside
Beat butter and sugar using a stand mixer until light and fluffy
Add egg and beat until combined
In a large bowl, whisk together flour, baking powder and cinnamon
Add four mixture and apple mixture to butter mixture alternatively, starting and ending with flour mixture
Beat on low just until blended
Transfer contents to a piping bag or gallon ziploc bag and snip the end
Pipe batter into greased doughnut pan
Bake at 350 F for 18-20 minutes, or until golden brown and springy to the touch
Combine 1 cup sugar, 1/2 tsp cinnamon and 1/2 tsp pumpkin spice in a bowl
Melt 1/4 cup butter in a small bowl
Once doughnuts are removed from oven and slightly cooled, dip one side of doughnuts quickly into butter, then dip into sugar mixture to coat
Allow butter to dry before serving
Enjoy!