Preheat your oven to 375 degrees Fahrenheit (190 degrees Celsius) and thoroughly butter two 6-cup donut pans.

In a saucepan, pour in the apple cider and heat over medium. Allow it to simmer, watching closely, until the cider is reduced to half a cup. Should it reduce excessively, simply add enough water to return to the required volume. Set this aside for later use.

In a large mixing bowl, combine the all-purpose flour, baking powder, baking soda, salt, 1 teaspoon of cinnamon, cardamom, and nutmeg. Whisk these ingredients until they are well blended and then set aside.

In another bowl, whisk together the white sugar, brown sugar, warm milk, 2 tablespoons of melted butter, vanilla extract, and the egg until smooth. Incorporate the previously prepared apple cider reduction along with the dry ingredients, mixing until just combined. Avoid overmixing; the batter should be slightly thick.

Using a spoon or piping bag, fill the prepared donut pans approximately three-quarters full with the batter.

Place in the center of the preheated oven and bake until the tops are lightly browned and the donuts spring back when gently touched, which should take about 10 to 12 minutes. Allow them to cool in the pans for 10 minutes before transferring them to a baking sheet lined with a silicone mat. Cut out any donut holes as necessary.

If desired, while the donuts are still warm, lightly brush the remaining melted butter over them.