In a medium saucepan over low heat, reduce the apple cider to about 3⁄4 cup, about 30 minutes. Set aside and cool to room temperature.
Sift together the flour, baking powder, baking soda, salt, and spices into a medium bowl and set aside. In a large bowl, use a stand mixer to cream together the butter granulated sugar, and brown sugar. Add the eggs one at a time and continue to mix until incorporated. Use a rubber spatula to occasionally scrape the sides and bottom of the bowl. Add the reserved reduced apple cider, buttermilk, and vanilla extract, mixing to combine. Add the flour mixture and combine. Gently fold the chopped apples into the batter.
Line a baking sheet with parchment paper and set aside.
Turn the dough out onto a well-floured surface. This dough can be sticky. Use a plastic bench scraper to fold the dough over and flour before cutting doughnuts.
Use a rolling pin to roll out the dough to 1⁄ 2 inch thickness. Lightly flour a doughnut cutter and use it to cut out the doughnuts, then place them onto the prepared baking sheet.
Line a baking sheet with several layers of paper towels and set aside.
In a large heavy-duty pot, add about 3-inches of canola oil. Heat to a temperature of 375 degrees F. Drop about 4 to 5 doughnuts into the oil, making sure not to overcrowd the pot. Cook for 1 to 2 minutes on each side, or until lightly golden brown.
Remove the doughnuts from the oil and allow to drain on the paper towels. In a small bowl, combine all of the ingredients for the cinnamon sugar; mix well. While the doughnuts are still warm, sprinkle them with the cinnamon sugar.