Preheat the oven to 375°F (190°C). Grease two 6-cup donut pans.
In a saucepan, simmer the apple cider over medium heat until reduced to 1/2 cup. If needed, add water to reach 1/2 cup. Set aside.
In a large bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, cardamom, and nutmeg. Set aside.
In another bowl, mix white sugar, brown sugar, milk, 2 tablespoons of melted butter, vanilla extract, and egg. Add the apple cider reduction and dry ingredients; whisk until you have a slightly thick batter (do not overmix).
Spoon or pipe the batter into the prepared donut pans, filling them about 3/4 full.
Bake in the center of the oven for 10-12 minutes until the tops are lightly browned and the donuts are slightly firm to touch.
Let the donuts cool in the pans for 10 minutes before transferring them to a sheet pan with a silicone baking mat. Cut out any donut holes as needed.
If desired, while still warm, brush the donuts lightly with the remaining melted butter.
To make the coating: Mix sugar and cinnamon in a shallow dish. Toss donuts in the mixture to coat.
Let the donuts cool completely before serving.
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