Preheat the oven to 350° F. Grease a 6-cup doughnut pan or 12-cup muffin pan with melted butter.

Bring the apple cider to a boil over high heat. Reduce the heat and simmer for 10-15 minutes or until reduced to about 1/2 cup. Remove from the heat and let cool.

In a large mixing bowl, stir together 1/2 cup boiled cider, melted butter, eggs, apple butter, vanilla, and brown sugar until combined. Add the flour, baking powder, baking soda, cinnamon, nutmeg, ginger, and salt, and mix until just combined. Toss the apples with the cinnamon sugar mix, then stir the apples into the batter.

Divide the batter evenly among the prepared pan, filling 1/2-2/3 of the way full. Bake for 12-13 minutes, until just set. Remove and let cool for 5 minutes, then run a knife around the edges to release and invert the pan.

Meanwhile, make the glaze. Add the butter to a small pot set over medium heat. Allow the butter to brown slightly until it smells toasted, about 2-3 minutes. Remove from the heat and whisk in the maple syrup, powdered sugar, instant coffee, and cinnamon. Dip or drizzle the doughnuts into the glaze.