Place cider in a sauce pan over medium heat until it comes to a low boil. Reduce heat to low and continue to simmer until reduced to 1 cup, about 20 minutes or so.
Preheat oven to 350 degrees.
In a medium size mixing bowl, whisk the flour, brown sugar, baking powder, baking soda, salt, cinnamon, nutmeg, and cardamom until combined. Set aside.
In a stand mixer, beat the cooled apple cider, eggs, melted butter, and vanilla extract until smooth. Slowly add the flour mixture and mix until completely combined.
Lightly grease your muffin pan of choice. Using a piping bag, fill each well on your pan just over half way full. Be sure to not over fill. This will cause the hole in the center to close up once the dough has risen and baked.
Bake for 10-13 minutes or until the donuts are light golden brown and spring back when lightly touched. Turn out of pan onto a cooling rack to cool.
Once donuts are completely cooled, melt butter and set aside. Combine granulated sugar and cinnamon in a small bowl.
Dunk the donuts in the melted butter or brush them with the melted butter. Roll all sides in the cinnamon sugar mixture. Set back onto the cooling rack until all the donuts are done.
These donuts are best fresh, but can be stored in an airtight container for 2 days or in the fridge for 4 days. Just pop in the microwave for 10 seconds to warm.