Core and chop the apples with the peel on. In a medium saucepan, combine apples, apple cider, and cinnamon sticks. Cook covered over medium heat until softened, about 5 minutes, then uncover and reduce for another 5 minutes. Remove cinnamon sticks and puree until smooth. Reduce further if needed to yield 1 cup. Let cool slightly.
In a large bowl, whisk together all-purpose flour, baking powder, baking soda, salt, cinnamon, and nutmeg. Set aside.
In a separate bowl, whisk coconut milk, apple cider vinegar, maple syrup, and vanilla extract. Set aside.
Using a hand or stand mixer with a paddle attachment, beat sugar and vegan shortening until creamy, about 2 minutes. Gradually mix in the cooled apple sauce, scraping down the bowl as needed.
On low speed, add half the flour mixture, then all the wet ingredients, then the remaining flour mixture. Mix until combined into a sticky dough, about 2 minutes.
Line a quarter sheet pan with parchment paper and dust with flour. Transfer the dough and shape into a rectangle about ½ inch thick. Wrap with plastic wrap and refrigerate for at least 2 hours or overnight.
Heat 2 inches of neutral oil in a deep, heavy-bottomed pot to 350°F. Place a cooling rack over a baking sheet. Mix sugar and cinnamon for the coating in a small bowl.
Flour a doughnut cutter and cut out donuts and centers, reworking scraps as needed.
Once the dough has chilled, fill a heavy-bottomed pot with about 2 inches of oil and heat over medium-high to 350˚F using a candy or deep-fry thermometer to measure temperature. Place a cooling rack over a sheet pan.
Fry donuts in small batches for about 2 minutes per side, maintaining 350°F. Transfer to the cooling rack to drain.
While still warm, roll donuts in the cinnamon sugar mixture and serve.