In a small saucepan over high heat, heat the apple cider. Bring to a boil and allow the cider to reduce by half. Set aside to cool.
In a mixing bowl, cream together butter and sugar.
Then, add in the egg, cider and milk. Beat until smooth. Set aside.
In a separate bowl, combine the flour, xanthan gum (if using), baking powder, baking soda, cinnamon, nutmeg, and salt.
Stir half of the dry ingredients into the wet ingredients.
Fold in the diced apple and then stir in the remaining flour mixture.
Flour a square of parchment paper.
Place the dough in the center of the parchment paper. Flour the top of the dough and using your hands pat it down into a disk about 1/2 inch thick. Wrap it in the parchment paper and put it in the freezer for 10-15 to make it firm.
Preheat oven to 425 degrees Fahrenheit.
Take the dough out of the freezer and cut either donut shapes or donut hole shapes using a cookie cutter or biscuit cutter.
Transfer each donut or donut hole to baking sheet lined with parchment paper. Bake for 10 minutes.
In the meantime, in a small bowl, combine cinnamon and sugar.
When the donuts are out of the oven, dip them in the bowl with the melted butter and the roll them in the cinnamon sugar mixture. Set on a wire rack to dry and cool.
Store in an airtight container. They will be best the day that they are made.