In a medium saucepan, bring the apple cider to a boil over medium heat. Once boiling, reduce to a simmer and cook for about 15-20 minutes, or until the cider has reduced to 1 cup. Let it cool completely before using.

In a large mixing bowl, combine the granulated sugar, melted butter, and vanilla extract. Add the eggs and the cooled apple cider. Mix until everything is well blended and smooth.

In another bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt. Gradually add the dry mixture to the wet ingredients, stirring until just combined. Be careful not to overmix.

Wrap the dough in plastic wrap and let it chill in the refrigerator for about 1 hour.

Once chilled, roll out the dough on a lightly floured surface to about 1/2-inch thickness. Use a donut cutter or two round cutters (one large and one small) to cut out donut shapes. Place the cutouts on a baking sheet lined with parchment paper.

In a deep pan, heat about 2 inches of oil to 350°F. Carefully add the donuts to the hot oil, frying a few at a time. Cook for about 2 minutes on each side, or until they are golden brown.

Remove from the oil and drain on paper towels.

In a small bowl, mix together the granulated sugar and ground cinnamon. While the donuts are still warm, roll them in the cinnamon-sugar mixture until fully coated.