Preheat your oven to 375 degrees Fahrenheit.
Sift together 3 cups flour, 1 tsp baking powder, 1 tsp cinnamon, 1/2 tsp baking soda, 1/2 tsp salt, and 1/2 tsp nutmeg. Set aside.
Using an electric hand mixer, cream together 2/3 cup butter and 1/2 cup of each sugar until light and fluffy.
Add the 2 eggs and 1 tsp vanilla and beat again until combined. Then add 1/2 cup applesauce and mix well again.
Add the sifted ingredients into the applesauce mixture and mix well with a wooden spoon or rubber spatula. The dough will be sticky.
Using a #24 cookie scoop (2.5 - 3 Tbsp), drop the cookie dough onto parchment paper-lined cookie sheets. Use your hand to press the dough ball down to about ½” thick.
Bake for 9-11 minutes or until the bottoms have browned slightly and the edges have set.
Move immediately to a cooling rack. While the cookies are cooling, make the frosting.