Preheat your oven to 350°F (175°C) and lightly grease your 8x8-inch square baking pan or line it with parchment paper for easy removal.
In a large mixing bowl, thoroughly whisk together the all-purpose flour, oat flour, cocoa powder, sugar, salt, baking powder, and baking soda until completely uniform and lump-free.
Add the vanilla extract, oil, and milk straight into your dry flour ingredients.
Stir gently with a flexible spatula or whisk just until a smooth cake batter forms and all large flour streaks disappear.
Pour the smooth chocolate batter directly into your prepared 8x8 pan, using your spatula to smooth it evenly into the corners.
Place the pan onto the center rack of your oven and bake at 350°F (175°C) for twenty-five to twenty-eight minutes or until a toothpick inserted into the center comes out completely clean.
Remove from the oven and let the cake cool completely in the pan on a wire rack.
In a medium bowl, beat the two tablespoons of softened butter, powdered sugar, cocoa powder, vanilla extract, and one tablespoon of milk together until smooth.
Add the remaining tablespoon of milk a little at a time only if needed until the frosting reaches a thick, silky, and easily spreadable consistency.
Spread the frosting evenly over the top of the completely cooled cake before slicing.