Preheat oven to 350 degrees F. Pour apple cider into a pan and with heat on high, boil cider until reduced in size to 2/3 cup (about 10 minutes). (You can lower the heat as it boils down.) Let cool.
In a medium-sized bowl, whisk together dry ingredients: flour, baking powder, salt, ground cinnamon, and ground ginger. Set aside.
In a large bow, mix together cooled cider, brown sugar, coconut oil, milk, and the egg until thoroughly mixed. Add in dry ingredients and mix until just incorporated.
You can either use a piping bag to pipe the batter into the pans or spoon the batter evenly into the pans. This batter is a bit more runny so it can be a little messy! Bake for about 10 minutes or until doughnuts spring back slightly when touched. Let cool in pan for 10 minutes then remove onto wire rack to cool. Coat with butter/cinnamon sugar topping.
Place melted butter in a bowl. In a separate medium bowl, mix together sugar and cinnamon until combined. Dip doughnuts in melted butter, then cinnamon sugar mixture. Return to cooling rack. Best eaten day-of but can be stored in airtight container.