Add your apple cider to a saucepan, and bring it to a simmer. Leave it on medium/low heat for 20 minutes to reduce the liquid. Stir occasionally.
While the cider simmers, add the flour, salt, baking powder, baking soda, cinnamon, nutmeg, and cardamom to a large bowl. Whisk to combine evenly.
Once the cider is reduced, add it to a medium-sized bowl, along with the starter and apple butter. Whisk to combine.
Then, add the egg, sugars, melted butter, and vanilla to the cider mixture. Whisk thoroughly.
Add the wet ingredients to the dry. Using a silicone spatula, fold the ingredients together until the flour is fully absorbed. Be careful not to over-mix!
Sprinkle a parchment-lined baking sheet with flour. Pour the dough on the floured surface, and press it into an even rectangle, roughly 1" thick. Cover with plastic wrap and refrigerate for at least one hour (up to 24 hours).
After the cold rest, heat your oil to 350°F. The oil should be roughly 3" deep in the pan.
Using a donut cutter, cut out your shapes. Re-flatten the dough as needed to use it all up!
Fry the donuts for roughly 2 minutes on each side (until golden brown).
While still warm, coat the donuts in cinnamon sugar, and ENJOY!