Line an 18x26-inch baking sheet with plastic wrap.
On a cutting board with a sharp knife, core, optionally peel, and coarsely chop the apples.
In a saucepan over medium heat, add the chopped apples and 1 1/2 cups of the apple cider.
Cover the saucepan and cook, while stirring often, until the apples begin to soften, about 8-10 minutes.
Uncover the apples and cook until the cider is reduced, about 15 minutes.
With an immersion blender or with the apple mixture transferred into a blender, purée until smooth.
In the same pan over medium heat, add the puréed apple mixture and cook, while stirring often, until the mixture is reduced to 1 cup, about 10 minutes.
Transfer the apple mixture from the stovetop to a wire rack and let it cool slightly.
In a large bowl, add 3 1/2 cups of the flour, the baking powder, 1 1/2 teaspoons of the cinnamon, the salt, the baking soda, and the nutmeg and whisk to combine. Set the flour mixture aside.
In a separate large bowl with an electric mixer on medium speed, add 2/3 cup of the granulated sugar and the vegetable shortening and beat until the sugar mixture has a sandy appearance.
Add the egg and the egg yolk to the sugar mixture and beat to combine.
Add the apple mixture to the sugar-egg mixture and beat to combine, scraping the sides of the bowl as needed.
Add 1/2 of the flour mixture to the apple mixture and beat to combine.
Add the buttermilk and the vanilla extract and beat to combine.
Add the remaining flour mixture to the dough mixture and beat until just combined, taking care not to overmix.
Transfer the dough mixture to the center of the plastic wrap on the prepared baking sheet.
With slightly floured or wet hands, pat the dough mixture into a 7x11-inch rectangle. The dough will be sticky.
Wrap the dough mixture in the plastic wrap and transfer the baking sheet to the refrigerator to chill for at least 8 hours and up to overnight.
When ready to prepare the donuts, line a clean 18x26-inch baking sheet with paper towels.
In a deep pot or tall-sided skillet, add about 2 inches of the vegetable oil.
Heat the oil to 350 degrees F.
In a small pot over medium heat, add the remaining apple cider and cook until reduced to about 1/4 cup, about 7-8 minutes.
Add the confectioners' sugar to the reduced apple cider and whisk until smooth.
Cover the cider glaze to keep it warm and set it aside.
In a shallow bowl, add the remaining granulated sugar and the remaining cinnamon and mix to combine. Set the cinnamon-sugar aside.
Generously dust a clean, flat surface and a rolling pin with the extra flour.
Turn the dough mixture onto the floured surface and roll it into a 9x12-inch rectangle.
With a 3-inch round cutter, cut the dough into 12 donuts.
With a 1-inch round cutter, cut out the center of each donut.
When the vegetable oil has reached the desired temperature, add about 2-3 donuts to the heated oil, and fry, flipping halfway through, until the donuts are golden-brown on each side and cooked through, about 2-3 minutes per side.
Transfer the fried donuts to the prepared baking sheet.
Complete the frying process with the remaining donuts and the donut holes.
When the fried donuts are cool enough to handle but still warm, dip the tops of the donuts into the glaze mixture and then immediately into the cinnamon-sugar mixture.
When the fried donut holes are cool enough to handle, fully submerse the donut holes into the glaze mixture and immediately roll and completely coat them with the cinnamon-sugar mixture.
Plate the donuts and serve warm.