Put your apple cider in a sauce pan and heat it over medium. Bring it to a boil, then reduce to a simmer, stirring every few minutes, until it has reduced to about ⅓ cop worth. This should take about 20 minutes, but can be made the day before.
Preheat your oven to 350℉. Grease your donut pans.
In a mixing bowl, sift together the flour, baking powder, baking soda, cinnamon, pumpkin pie spice, and salt.
In another bowl, whisk the egg and sugars together until combined. Add in the cider, buttermilk, and vanilla, mixing well.
Pour the wet mixture into the dry mix and stir until it starts to come together. Try not to over-mix or the donut will be extra dense.
Scoop the mix into a piping bag and pipe into your donut pan, filling each mold about halfway. You can also just pour or use a spoon if you prefer, I just find piping easier.
Bake for about 10 minutes, or until a toothpick inserted into a donut comes out clean.
Let cool for about 5 minutes, then pull out and mostly finish cooling on a cooling rack.
Meanwhile, melt the butter, and combine the white sugar, cinnamon, and nutmeg.
Brush the donuts with butter, then dip them in the cinnamon-sugar mix. Enjoy while warm for the tastiest donuts!