Core and coarsely chop the apples. I've seen both peeled and unpeeled and I prefer peeled.  Combine the apples with 1 1/2 cups apple cider in a medium saucepan. Cover and cook for about 8 minutes or until softened.

Uncover then continue cooking until the apples are tender and the cider is almost completely reduced (about 5 minutes).

Puree with an immersion blender or a food processor until smooth. You should have about 1 cup of the sauce in the end (boil down and reduce further if you have too much). Let cool slightly.

Meanwhile, in a bowl combine the flour, baking powder, baking soda, cinnamon, salt, and nutmeg. Set aside.

Using an electric mixer, beat the butter, vanilla extract, and sugar until smooth. Add eggs one at a time and beat until completely incorporated.  Use a spatula to scrape down the sides of the mixing bowl to make sure the mixture is even

Slowly mix in the reduced apple cider and the buttermilk until it's just combined. Slowly add the flour mixture until the dough comes together. Be careful of over mixing.

Line a baking sheet with parchment paper and sprinkle with flour.

Turn the dough onto one of the sheets and sprinkle the top with four.

Flatten the dough with your hands until it's about 1/2" thick. Use flour if the dough is still wet.

Cover with plastic wrap then transfer the dough to the freezer until it is slightly hardened (about 20 minutes) or refrigerate overnight.

Remove the dough from the freezer or refrigerator. Using a 3 1/2" round cutter and a 1" cutter for the center hole, cut out the donut shapes. Place the cut donuts and holes onto the second baking sheet. Combine remaining dough and scraps, refrigerate briefly, then repeat cutting until there is no dough left.

Refrigerate doughnuts again for 20-30 minutes.

Turn the deep fryer on to 350°F.  If you haven't a deep fryer, carefully set up on the stove in a deep, thick bottomed pot.

Add at least 3" of oil to a deep or medium-deep pan. Use a candy thermometer to measure the temperature and heat the oil until it's at 350°F.

Add a few doughnuts at a time to the oil while not crowding the pan. Fry until golden brown (about 60 seconds) then turn until the other side is golden (about 30 to 60 seconds). Check the time with one doughnut to be sure it is done inside. Don't forget to fry the doughnut holes as your tribute for hard work. Drain on wire screen. I love golden brown and delicious but cake doughnuts can be tricky. Too dark and they may taste burnt even if they aren't. All the extra sugar from the apples and apple cider will brown fast so keeping the fat hot by frying only a few at at a time is important.

Dip the top of the warm donuts into the glaze. 

 Or, coat them with the cinnamon sugar mix. Place the still warm doughnuts into the sugar and cover them by burying them in the cinnamon sugar with a soup spoon. Cool doughnuts will not hold the cinnamon sugar so it's important to coat them in sugar while they are hot.

Reduce 1 cup apple cider in a small saucepan until you only have about 1/4 cup left. Whisk in the confectioner's sugar until smooth and glossy. Set aside.