Preheat: Heat the oven to 350°F. Brush a 6-cavity donut pan or 12-cup muffin pan with melted butter.
Reduce the cider: Boil the apple cider over high heat, then lower the heat and simmer 10 to 15 minutes, until reduced to about ½ cup. Let cool.
Mix wet ingredients: In a large bowl, stir together ½ cup boiled cider, melted butter, eggs, apple butter, vanilla, and brown sugar.
Add dry ingredients: Add the flour, baking powder, baking soda, cinnamon, nutmeg, ginger, and salt. Stir just until combined.
Add apples: Toss the chopped apple with cinnamon sugar, then fold into the batter.
Fill: Fill each donut cavity ½ to ⅔ full.
Bake: Bake 12 to 13 minutes, until the donuts spring back when touched.
Cool: Cool in the pan 5 minutes, loosen the edges with a knife, and turn out. Repeat with the remaining batter.
Make the glaze: Melt the butter in a small pot over medium heat and cook 2 to 3 minutes, until golden and nutty. Remove from heat and whisk in the maple syrup, powdered sugar, instant coffee, and cinnamon.
Glaze: Dip or drizzle the donuts with glaze. Best enjoyed warm.