Pour the apple cider into a small saucepan and simmer over medium heat until reduced to 3/4 cup, about 15–18 minutes. It will look syrupy and smell intensely apple-y. Set aside to cool completely. Reserve 2 tablespoons for the glaze and 2 tablespoons for brushing; the rest goes into the batter.
Heat the oven to 350°F (175°C). Grease a 10-cup bundt pan generously with butter, then dust it with flour, tapping out the excess. Get into every groove — this cake needs to release cleanly.
Whisk the flour, baking powder, baking soda, cinnamon, nutmeg, cloves, and salt in a medium bowl.
Beat the butter, granulated sugar, and brown sugar in a large bowl on medium-high speed until light and fluffy, about 3 minutes. Beat in the eggs one at a time, then the vanilla.
Beat in 1/2 cup of the cooled reduced cider and the applesauce on low speed until just combined. The batter may look slightly curdled — that is fine.
Add the dry ingredients in two additions, mixing on low just until no dry streaks remain. Fold in the grated apple with a spatula. Do not overmix, or the crumb turns tough.
Scrape the batter into the prepared bundt pan and smooth the top. Bake at 350°F (175°C) for 45–50 minutes, until a skewer inserted into the thickest part comes out with just a few moist crumbs.
Cool the cake in the pan on a rack for 15 minutes, then invert it onto the rack and let it cool completely, about 1 hour.
While the cake is still just barely warm, brush it all over with the 2 tablespoons of reduced cider you reserved for brushing, then sprinkle the cinnamon-sugar coating all over the top and sides, pressing gently so it sticks.
Whisk the powdered sugar with 2 tablespoons of the reduced cider until smooth and drizzle it over the cooled cake. Slice and serve at room temperature.