Preheat the oven to 350 degrees and grease a baking sheet. In a bowl, combine all the dry ingredients except the sugar and mix well.
In a separate bowl, cream together the butter, both sugars, and the vanilla. Add the applesauce to the sugar bowl and mix well again.
Combine both the dry and wet ingredients and mix well until a thick dough forms.
Roll into 1-inch balls and place on the prepared baking sheet.
Press down the balls with a fork in a criss-cross pattern. Sprinkle with coarse sugar and bake for about 8-10 minutes or until the edges of the bottoms start to turn a very light gold color. Leave on the pan to cool for a few minutes.
If you want to cut shapes out of the dough, once mixed, cover it and chill it for a couple of hours. Then, place the dough in between two sheets of wax paper and roll it out, making sure to use plenty of flour if need be so it doesn’t stick. Cut out your shapes and bake them following the same baking time and temps.