Whisk flour, cinnamon, nutmeg, baking powder, baking soda, and salt together in a bowl.
In another large bowl, whisk together the apple cider, sugar, buttermilk, melted butter, and egg.
Whisk half of the flour mixture into the apple cider mixture until it becomes smooth with no clumps.
Add the remainder of the flour mixture and, using a rubber or silicone spatula, fold the dough together until the mixture has no clumps remaining. Heavily dust a working space on your counter with flour. Turn dough out onto work surface and then dust the top of the dough with additional flour.
Using your hands, gently pat dough into 1/3 inch thick round about 10 to 12 inches in diameter.
Now, using a floured 3 inch round cutter, cut about 10 rounds.
Using a 1 inch round cutter, cut out the centers of each round to form your doughnuts and doughnut holes.
Lightly dust a baking sheet with flour and transfer doughnut rounds and holes onto the tray.
Combine your dough scraps and knead together into a ball. Pat down again into a 1/3 inch thick round and cut out the rest of your doughnuts and rounds.
Transfer them to the sheet tray and refrigerate while you heat your oil.
Set a wire rack on top of another sheet tray and place 3 layers of paper towels over the top of half of the rack.
Add your oil to a large dutch oven or cast iron skillet. (You can also obviously use a fryer if you happen to have one.) make sure your oil is at least an inch and a half deep. Heat the dutch oven to 350°f on a medium heat.