In the bowl of a stand mixer fitted with the paddle attachment, cream the butter and sugars (both granulated and brown) on medium-high until light and fluffy (about 2-3 minutes).
With the mixer on low, add the eggs one at a time, mixing until just incorporated.
Combine the dry ingredients in one bowl (flour, baking powder, salt, cinnamon and nutmeg) and the remaining wet ingredients (vanilla extract, apple cider and apple sauce) in another.
With the mixer on low, add the dry ingredients in two equal parts, alternating with the wet ingredients in one part. Mix until combined.
Pour the batter into a well-greased 6-cup bundt pan and bake at 350F for 35-40 minutes.
Remove the cake from the oven and allow to cool until the pan is no longer hot. Turn out onto a cooling rack.
Once the cake has cooled completely, stir together the remaining granulated sugar and cinnamon.
Brush with the melted butter and sprinkle liberally with the cinnamon sugar.
Notes
Testing for doneness: The cake is done when a toothpick comes out with a just few cooked crumbs.
Easy pan removal: The trick to easily removing a cake from a bundt pan starts in the preparation. You'll want to liberally grease the pan and I've found a homemade pan release made from equal parts oil, flour and vegetable shortening works best. Once the pan is cool enough to handle, turn the cake out.
Storage: This cake can be stored at room temperature and is best when eaten within 2-3 days.
Freezing: Wrap in plastic wrap and aluminum foil prior to coating with butter, cinnamon and sugar. Freeze for up to 3 months.