In a small saucepan, bring apple cider to a simmer over medium heat. Cook, stirring occasionally, until reduced to about ½ cup, 15–20 minutes. Set aside to cool slightly.
Preheat oven to 350°F. Butter two standard doughnut pans (or one, working in batches).
In a medium bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, nutmeg, and cloves.
In a large bowl, whisk melted butter with brown sugar until smooth. Whisk in eggs one at a time, then vanilla, the reduced cider, and sour cream. Fold in the grated apple.
Add the dry ingredients to the wet and fold just until no streaks of flour remain. Do not overmix; batter will be thick.
Spoon batter into a piping bag or zip-top bag with the corner snipped off, and pipe into doughnut wells, filling each about two-thirds full.
Bake until doughnuts spring back when lightly pressed and a toothpick comes out clean, 9–11 minutes.
Let cool in the pan 5 minutes, then turn out onto a wire rack.
Melt butter for coating; stir together sugar and cinnamon for coating in a shallow bowl. While doughnuts are still warm, dip each side into melted butter, then dredge in cinnamon sugar to coat completely.