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Pour cider into a nonstick saucepan and heat over medium until it reduces to ⅓ cup, forming a thick syrup. Set aside to cool.
In a large bowl, combine egg, granulated sugar, brown sugar, butter, milk, and vanilla extract. Mix until smooth and set aside.
In a separate bowl, whisk together flour, baking powder, baking soda, cinnamon, nutmeg, and cardamom.
Gradually add the dry mixture to the wet mixture, along with the reduced cider, and stir until just combined. Avoid over-mixing.
Pour the batter into a greased donut pan, filling each mold halfway. Tip: Using a piping bag can make this step easier and cleaner.
Bake at 350°F for approximately 10 minutes or until a toothpick inserted into a donut comes out clean.
Allow the donuts to cool in the pan for 5 minutes, then remove them.
While the donuts are cooling, mix sugar, cinnamon, nutmeg, and cardamom in a wide bowl.
Brush each donut with melted butter and roll them in the sugar mixture until fully coated.
Serve immediately and enjoy!