Add apple cider to a medium, saucepan over low heat. Let apple cider simmer for about 20-25 minutes until reduced, (you should have about 1 1/2 cups). Remove from heat to let cool.
In a medium bowl sift flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
In a large bowl, using a hand mixer cream together the butter, granulated sugar, and brown sugar until combined. Add eggs one at a time and mix until combined. Add in 1 cup reduced apple cider, buttermilk, and vanilla and mix.
Add the flour mixture a mix until combined.
Gently fold in the chopped apples with a spatula just until combined.
Line a baking sheet with parchment paper and set aside.
Turn half the dough out onto a well-floured surface, this dough is sticky. Gently roll the dough around until it becomes less sticky.
Using a rolling pin, roll out the dough to about 1/2 inch thickness. Using a lightly floured donut cutter, cut out the donuts and place them on the baking sheet. Scoop up the dough scraps and repeat. Then repeat with the other half of the dough.
Place a cooling rack on a baking sheet or over some paper towels.
Add about 3 inches of oil into a heavy bottom pot.
Check your oil with a thermometor that it has reached 325 degrees.
Drop 3-4 donuts in the pot at a time (you don't want to crowd them) Cook for 1-2 minutes then flip and cook another 1-2 minutes. Remove the donuts and place them on the wire rack to cool.
Mix powdered sugar and remaining apple cider in a small bowl until combined. Dip the donuts in glaze and place them back on the cooling rack.