To make the cinnamon sugar: whisk together all the ingredients. Use immediately, or store in an airtight container for up to 2 weeks.
To make the doughnuts: boil the cider in a small saucepan for about 5 minutes, letting it bubble away until reduced to a syrupy 2 tablespoons. Set aside to cool.
Meanwhile, in the bowl of a stand mixer fitted with the paddle attachment, cream together the butter and sugar. Drop in the egg and mix well, and then add the flour, baking powder, baking soda, cinnamon, nutmeg, and salt (I recommend mixing the dry ingredients first in another bowl and then adding them so the ingredients are evenly distributed without over-mixing the dough). Mix until just combined, scraping down the sides of the bowl with a spatula every so often. Drizzle in the reduced cider and the buttermilk, and mix until just combined. The dough should hold together, but still be on the sticky side.
Scrape the dough onto a sheet of parchment paper and put another sheet of paper on top. Using a rolling pin, flatten and roll the dough until it’s 3/8 to 1/2-inch thick. Put the dough in the refrigerator and chill for 30 minutes to 1 hour, until it no longer clings when you try to peel off the parchment paper (I chilled mine longer).
Heat at least 2 inches of oil in a heavy-bottomed pot fitted with a deep-fry thermometer until the oil reaches 350 degrees F. Prepare two baking sheets: one lined with paper towels, and the other dusted with flour.
Peel off the top sheet of parchment paper and flip the dough onto a floured work surface. Remove the second sheet of parchment paper and dust the dough with flour. Using a floured biscuit or doughnut cutter (or other round cutters), cut the dough into 2 1/2-to-3-inch tubular rounds (with about 1-to-1 1/2-inch holes in the center depending on the doughnut’s diameter) and put them on the flour-dusted baking sheet.
Fry the doughnuts in batches, taking care not to crowd the pot, until golden brown all over, 1 to 2 minutes per side. Drain on the paper towel-lined baking sheet.
Toss them in the cinnamon sugar while they are still warm.