Pour the apple cider into a saucepan. Over medium heat, let the cider simmer until it's reduced to 1/2 cup. Set aside to cool for 30 minutes, or make it a day or two in advance.
Preheat the oven to 375, and spray two standard doughnut pans (6 doughnuts each) with nonstick spray. If you only have one pan, you can just bake half the batter at a time.
In a bowl, whisk together the cooled reduced cider with the applesauce, melted butter, egg, vanilla and brown sugar.
In a separate bowl, sift together the flour, spices, baking soda and salt. Add the dry ingredients to the wet and whisk until combined.
Either spoon the batter into the greased pans, or fill a piping bag, snip off the end, and pipe the batter into the pans.
Bake the doughnuts on the center oven rack until risen and cooked through, about 11-12 minutes. Let cool in the pans for 4-5 minutes, then gently turn out onto a cooling rack.
To coat the doughnuts, dip a pastry brush in the melted butter and lightly brush the warm doughnuts with the butter. Coat in the cinnamon/sugar mixture.