Preheat oven to 325 degrees F (165 degrees C). Lightly grease a doughnut pan.
In a large bowl, mix flour, sugar, baking powder, apple pie spice, and salt. Stir in cider, milk, eggs, vanilla, and melted butter. Beat together until well blended.
Fill each doughnut cup ¾ full.
In a medium pan, heat the apple cider until reduced by half.
Add in brown sugar and milk over medium heat until it comes to a boil. Boil for 3-4 minutes, then remove from the heat and stir in the butter until melted.
Whisk in the powdered sugar until smooth.
Working quickly, dip the tops of each donut in the glaze before inverting onto a wire cooling rack.
If the glaze firms up too quickly, simply reheat over low on the stove until liquid, then remove from the heat and continue glazing.