Preheat oven to 350°F, and set 2 of the oven racks to be as close to the center of the oven as possible. Line 2 standard baking sheets with parchment paper.
In a large bowl or in the bowl of a stand mixer, add the butter, brown sugar, and cane sugar. Using a hand mixer or the stand mixer, cream the butter and sugar on high for about 30-40 seconds.
Add to the bowl, the applesauce and peanut butter, and mix on high again until fully combined.
Add the flour, baking powder, and salt, then mix on medium just until the dough comes together and thickens. If using, stir in the chocolate chips.
If rolling in sugar, pour the ¼ cup of cane sugar into a small bowl. Scoop the cookies, roll them in the sugar, and place them on the prepared baking sheets at least one inch apart. There should be about 10-12 cookies per tray. Using a fork, press down in a crisscross pattern until the cookies should be about a ½- to -¾-inch thick.
Bake the cookies for 10-12 minutes, swapping their positions in the oven after 5 minutes. Remove them from the oven when the edges are golden brown, and allow them to cool completely before removing from the pan to avoid crumbling.
Once completely cooled, enjoy the cookies! Store leftovers in a sealed container at room temperature or in the fridge. They may also be frozen for up to 1 month.