In a medium bowl, combine the oats, flour, baking soda, salt, and cinnamon. Set aside.
In a large bowl, mix butter, brown sugar, and granulated sugar, on medium speed until thoroughly mixed (about 1-2 minutes).
Add unsweetened applesauce and vanilla extract and beat on high until combined, about another minute.
Add in flour mixture and combine on low speed, being careful not to overmix.
Chill dough for 45 minutes.
Preheat the oven to 350℉. Line a cookie sheet with parchment paper. Set aside.
Scoop about 1 ½ Tablespoons of dough for each cookie, and place about 2-3 inches apart on the baking sheets.
Bake for 10-12 minutes until golden brown on the edges.
Remove from the oven and let sit on a baking sheet for at least 8-10 minutes.
Remove to a wire rack to finish cooling.
While the cookies are cooling, make the icing. Place the sifted confectioner’s sugar in a shallow bowl. Add vanilla and 1 Tablespoon of milk at a time, adding more until desired consistency is reached. The icing will be fairly thick.
When cookies are completely cooled, dip the tops of the cookies by turning them upside down and dip in the icing. Pull the cookie straight up, letting the excess icing drip back into the bowl. Turn the cookie right-side-up and transfer it back to the cooling rack. Let icing set for 15-30 minutes.