In a small saucepan, bring cider to boil over medium high heat for about 8 minutes until reduced to 1 cup. Set aside to cool completely.
In a large bowl combine flour, baking powder, baking soda, salt, nutmeg and cinnamon. Set aside.
In a separate large bowl, cream together eggs, sugars, melted butter, and cooled apple cider.
Pour the wet ingredients in with the dry ingredients and stir until just combined. (Dough will be sticky)
Cover the bowl with plastic wrap and place in the fridge to chill for one hour (up to overnight).
Sprinkle flour over a flat surface and roll out the dough to about ½ inch thick.
Use a doughnut cutter (or two different sized round cutters) to cut as many doughnuts as possible.
Fill a frying pan with 1 inch of oil heated to 325 degrees.
Allow the donuts to cook for about 1-2 minute then flip and fry for an additional 1 minute or so on the other side. Be careful not to let them burn.
Let them drain on paper towels to soak the excess grease.
Allow to cool for about 5 minutes before dipping in glaze or cinnamon sugar.
Mix all glaze ingredients in a bowl, and whisk until smooth. Carefully dip both sides of the doughnut into the glaze, or drizzle glaze over the top.
Place dipped donuts on a wire cooling rack over a sheet pan to catch any excess glaze.
Allow glaze to set for about 20 minutes and enjoy.