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In a small saucepan, bring cider to a boil and cook until reduced by about half. Remove from heat and cool completely.
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Whisk together flour, brown sugar, baking powder, salt, baking soda, cardamom, cinnamon, and nutmeg.
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In a separate bowl, whisk together eggs, butter, and cider. Stir into the dry ingredients. It will be sticky. Cover and refrigerate for 1 hour.
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Make the cinnamon sugar: stir together sugar and cinnamon.
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Divide dough in half on a well floured surface. Roll each portion to ½" thickness and cut with doughnut cutter.
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Heat oil to 365F. Fry doughnuts, a few at a time, for 1 - 1 ½ minutes on each side. Drain on paper towels. Toss hot doughnuts with cinnamon sugar mixture.