Add butter and powdered sugar to a bowl of a stand mixer. Mix on low (carefully, so the sugar doesn't go everywhere) until fluffy, about 2 minutes. Add egg and vanilla extract and mix until as combined as possible. In a separate bowl, mix all-purpose flour, almond flour, and salt together. Add to butter mixture and mix on low until dough forms. Set aside.
Preheat oven to 325 degrees. Spray a 9x9 square baking pan with nonstick spray and then line with parchment paper. Next, trace square pan onto another sheet of parchment paper to serve as a stencil for the top layer of cookie dough. Spread half of the cookie dough over the square stencil so that it's in an even layer. An offset spatula will help make this process easier. Spread the other half of the dough into the baking pan. Place both cookie layers in the freezer while you prep the center layer.
Mix all the ingredients together for brown sugar cinnamon layer in a small bowl until combined. You may need to smash the ingredients together with the back of a spoon to get the cinnamon to combine well into the brown sugar. The mixture will be sandy. Remove the cookie layers from the freezer. Sprinkle the brown sugar cinnamon over the bottom cookie layer sitting in the baking pan. Press the brown sugar cinnamon to compact it into one layer. Top the brown sugar layer with the top cookie layer and push down.
Bake in preheated oven for 30 minutes until the center is baked through. The edges will be slightly golden brown but the top will mostly be light. Remove from oven and let cool completely before putting icing on top, about 1 hour.
For the icing, mix all ingredients together with a whisk until smooth. The whisk will help remove any lumps from the powdered sugar. Pour over the cooled square of cookie bars while they are still in the pan. Let the icing harden before you slice into bars.