In a medium saucepan, combine the apples and apple cider and heat over medium heat until boiling. Cook, uncovered for 15-20 minutes until the apples have softened and the cider is mostly reduced with maybe ½ cup of the liquid left. Puree with an immersion blender or allow to cool before transferring to a blender to puree until smooth. You should end up with about 1 cup of concentrated, pureed apple. If you have more, the cider probably needs to reduce a bit more, so just return to the pan and cook down a bit more until you are left with just 1 cup of puree.
In a large bowl, whisk together the flour, baking powder, cinnamon, salt, baking soda, and nutmeg.
In a separate bowl, combine the sugar, butter, eggs, and reduced apple cider and apple puree. Whisk well, then add the buttermilk and vanilla and whisk again. Pour the liquid mixture into the dry ingredients and stir just to combine, taking care not to overmix.
Line a baking sheet with parchment paper and dust generously with flour. Turn the donut dough out onto the floured surface and sprinkle a little flour over the top. Dust your hands with flour and use them to pat the dough into a rectangle about ½-inch thick. The dough will be very sticky. Cover with plastic wrap and refrigerate for 2 hours (or stick in the freezer for 20 minutes).
When the dough has chilled, heat 2 inches of vegetable oil in a heavy pot or dutch oven until the temperature reaches 375 degrees F on a thermometer. Set a wire rack over a baking sheet lined with paper towels for draining the donuts after frying them. In a shallow bowl, combine the sugar and cinnamon for the topping and set aside.
Cut out the donuts using a donut cutter or a 3-inch biscuit cutter for the outer circle and a 1-inch cutter for the inside circle (I didn't have a 1-inch cutter so I used my largest frosting tip that was 1-inch in diameter). Save the inside circles for donuts holes and combine the rest of the dough and pat out again and cut into donuts until all the dough has been used.
Working in batches, fry 2 or 3 donuts at a time in the hot oil, cooking for 2-3 minutes per side until golden brown on the outside and cooked through. Transfer to the wire rack to cool, then dip into the cinnamon sugar mixture, turning to coat. Serve warm.
If you choose to make the apple cider glaze, just whisk the powdered sugar and apple cider together in a bowl, then dip the donuts into the glaze and let set for 15 minutes.