Reduce the Apple Cider: Pour 1 cup of apple cider into a saucepan. Bring to a simmer over medium heat and cook until reduced to 1/4 cup. This concentrates the apple flavor. Let it cool completely.
Preheat the Oven and Prepare the Pan: Preheat your oven to 350°F (175°C). Grease a donut pan very well with cooking spray.
Combine Dry Ingredients: In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, allspice, and salt.
Cream Butter and Sugars: In a large bowl, cream together the softened butter, granulated sugar, and brown sugar until light and fluffy.
Add Eggs and Vanilla: Beat in the eggs one at a time, then stir in the vanilla extract.
Alternate Adding Dry and Wet Ingredients: Gradually add the dry ingredients to the wet ingredients, alternating with the reduced apple cider and buttermilk, beginning and ending with the dry ingredients. Mix until just combined.
Fill the Donut Pan: Transfer the batter to a large zip-top bag or piping bag. Snip off the corner of the bag and pipe the batter into the prepared donut pan, filling each cavity about two-thirds full.
Bake the Donuts: Bake for 10-12 minutes, or until a toothpick inserted into the center comes out clean.
Cool and Unmold: Let the donuts cool in the pan for a few minutes before transferring them to a wire rack to cool completely.
Make the Cinnamon-Sugar Coating: In a small bowl, combine the granulated sugar and cinnamon for the coating. In another bowl, melt the butter.
Coat the Donuts: Dip each donut in the melted butter, then immediately dip it into the cinnamon-sugar mixture, coating evenly.