In a small saucepan, bring the apple cider to a boil over medium heat. Reduce the heat to a simmer and cook, uncovered, until the cider has reduced to about 1/3 cup and is syrupy, roughly 15 minutes. Remove from heat and let cool completely to room temperature.

In a large bowl, whisk together the egg, granulated sugar, brown sugar, melted butter, milk, and vanilla extract until smooth and well combined.

In a separate bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and cardamom.

Add the dry ingredients to the wet mixture, then pour in the cooled reduced cider. Stir gently with a spatula until just combined; be careful not to overmix. A few small lumps are fine.

Grease a standard donut pan. Transfer the batter to a piping bag or a zip-top bag with the corner snipped off, and pipe the batter into each cavity, filling about halfway full.

Bake in a preheated 350°F oven for 10 to 12 minutes, or until a toothpick inserted into a donut comes out clean. Let cool in the pan for 5 minutes, then turn out onto a wire rack.

While the donuts cool, mix the granulated sugar, cinnamon, nutmeg, and cardamom in a shallow bowl.

Brush each warm donut generously with melted butter, then roll in the spiced sugar mixture until fully coated. Serve immediately for the best texture.