Simmer the apple cider in a small saucepan until reduced to about 3/4 cup. Cool completely.
Heat oven to 350°F. Line a 12-cup muffin pan with paper liners.
Whisk flour, baking powder, baking soda, salt, 1 1/2 tsp cinnamon, and nutmeg in a bowl.
Cream 1/2 cup softened butter with granulated and brown sugars until light and fluffy, 2–3 minutes.
Beat in eggs one at a time, then mix in 1 1/2 tsp vanilla and applesauce.
Stir in 1/2 cup of the cooled reduced cider. Reserve the rest for frosting.
Add dry ingredients to the wet mixture and stir just until no dry streaks remain.
Divide batter among liners, filling each about two-thirds full.
Bake 17–19 minutes, until a toothpick comes out clean. Cool in pan 5 minutes, then cool completely on a rack.
Beat 3/4 cup softened butter until creamy. Gradually mix in powdered sugar, cinnamon, salt, and 1 tsp vanilla.
Drizzle in 2–3 tbsp of the remaining reduced cider until frosting is smooth and spreadable. Pipe or spread onto cooled cupcakes.