Preheat the oven to 350 degrees, set the wrack to the vertical center, and line two sheet pans with silicone baking mats or parchment paper.
Grind the oats into a course flour using a food processor (1-2 minutes on high). In a large bowl blend the oats, flour, baking soda, and salt. Set aside.
Cream the butter with the sugar until fluffy - about 2 minutes using an electric mixer on medium speed. Scrape down the sides of the mixer bowl, if needed, and add the egg and vanilla. Blend just enough to combine. Add the peanut butter and applesauce then mix 1-2 minutes until the batter becomes a uniform color and texture.
Slowly incorporate the dry ingredients adding about ⅓ at a time and mixing completely before adding the next ⅓.
Measure the batter in 1 ½ tablespoon portions (or use a #40 cookie scoop) on to the prepared baking sheets approximately two inches apart. Gently flatten each mound with a fork to make a cross hatch pattern on the tops.
Bake for 15 minutes until the edges are beginning to darken. If possible slide the parchment or silicon sheet, without removing the cookies, off the cookie sheets on to a cooling wrack immediately, otherwise let the cookies firm on the pan for a minute or two before transferring to a wrack to cool completely.
Notes
Optional: Sprinkle each cookie with a pinch of turbinado sugar before baking.