Pour the apple cider into a saucepan and bring to a boil. Reduce heat and simmer until it is reduced to 1 cup (about 20-25 minutes). Remove from heat and allow to cool to room temperature. (You can speed this up by placing it in the refrigerator.)
In a large mixing bowl, combine the flour, baking powder, baking soda, salt, cinnamon, nutmeg, granulated sugar and brown sugar.
In a separate bowl, combine the melted butter, eggs and reduced cider.
Blend the wet ingredients into the dry ingredients until just combined. Wrap the dough in plastic wrap and refrigerate for at least 1 hour.
Pour at least 1 inch of oil into a heavy bottomed pot or Dutch oven. Heat the oil to 325F degrees.
Prepare the cinnamon sugar mixture in a shallow bowl.
While the oil is heating, roll the dough out onto a lightly floured surface to 1/2-inch thick. Use a donut or biscuit cutter to cut the donut shapes.
Carefully lower the donuts into the oil using a slotted spatula or spoon. Fry until deeply golden brown on both sides, flipping as needed.
Remove donuts from the oil and allow the excess oil to drip off. Place into the cinnamon sugar mixture and coat on both sides.
Transfer donuts to a wire rack to cool. Enjoy slightly warm or at room temperature.