Place the rolled oats in a food processor, and process them until they are a coarse flour. Set aside.
Put the vegan buttery spread, brown sugar, and applesauce in a mixing bowl, and mix on medium speed until combined.
Add the oat flour, gluten free flour blend, cinnamon, salt, baking soda, and baking powder to the bowl. Mix on low speed until the dough comes together. It will be pretty thick.
Refrigerate the dough for one hour.
Preheat the oven to 350 degrees F. Line a cookie sheet with parchment paper.
Use a 1 ½ Tablespoon cookie dough scoop to scoop out the dough. Leave a fair amount of space between scoops on the cookie sheet. I could fit about 10 cookies on a cookie sheet. Bake the cookies for 10 minutes, then flatten the cookies with the back of a spoon. Bake for another 5 minutes.
Allow the cookies to cool full before icing. Repeat steps 6-7 with the remaining dough.
Once the cookies are cool, place the powdered sugar and water in a bowl. Beat on high speed until smooth. Add a little more water as needed - the texture should be a very thick glaze.
Take each cookie and gently dip the top into the icing. Don't fully submerge it, you want to see some cracks in the icing.
Set on the cookie sheets to allow the icing to harden. It will harden pretty quickly, in about 30 minutes or so.
Store cookies in an airtight container.