Preheat oven to 350 degrees F. and line a muffin pan with liners or grease.
In the bowl of a stand mixer, beat the butter and brown sugar together.
Add the applesauce, apple cider, vanilla, and eggs and beat until combined.
Add the flour, baking powder, baking soda, salt, cinnamon, nutmeg, allspice, and cloves and mix until just combined.
Scoop the batter into the prepared muffin pan and bake for 20-22 minutes, until a toothpick inserted in the middle comes out clean.
While the cupcakes cool, prepare the icing by beating the butter, cinnamon, nutmeg, and vanilla together.
Add the powdered sugar and apple cider and beat until fluffy.
Spread or pipe the buttercream onto the cooled cupcakes.