Preheat the oven to 350 degrees F ( 175 degrees C ). Lightly grease cookie sheets.
Stir applesauce, brown sugar, and white sugar together in a bowl. Sift flour, baking powder, salt, and cinnamon into a separate bowl; stir into applesauce mixture. Add rolled oats, raisins, and chopped nuts, if desired. Stir until combined. Drop spoonfuls of dough 2 inches apart onto the prepared baking sheets.
Bake in the preheated oven until edges are golden, about 10 minutes. Allow cookies to cool slightly before removing from the baking sheet.