To begin, mix together the dry ingredients in a medium bowl. That’s the flour, baking soda, baking powder, salt and spices. Set aside.
In a stand-up mixer, cream together the butter and sugar until the mixture is light and airy. Add the molasses and applesauce and combine. The dough may look a little curdled at this point. That’s okay! It will sort itself out once the rest of the ingredients are added.
Add the dry ingredients and mix to combine. Do not over mix that dough. Once you add the flour, mix it until it just starts coming together. At that point, turn it out onto a floured work surface and bring it together with your hands.
Transfer the dough to a piece of parchment paper, top with another paper and roll out to about ¼ inch thick. Refrigerate for 1 hour.
Preheat the oven to 400 degrees F. Once your dough is chilled, cut out shapes with your cookie cutters. Bake the cookies for 9-11 minutes, until browned on the edges and cooked all the way through. Allow the cookies to cool for 5 minutes on baking sheets before transferring to a wire rack to cool completely.
While cookies are cooling, whisk together sugar and milk until it is completely smooth. You may need to adjust the wet and dry ingredients. Make sure the frosting is pipeable but no so thin that it will lose its shape once on the cookie.