In the past, I practically always had these stains with a simple powdered sugar-water mixture, just not as badly. They came gradually and slowly got stronger. That's why I strongly assume, as Julia said, that the fat in the butter will destroy the cookie glaze. If you want to stick to a sugar-water mixture, it might be worth trying to glaze the cookies with raw egg white (unbeaten) before baking. The cookies then also shine slightly. Maybe the baked egg white doesn't let the fat through? I could imagine it anyway and I will try it once! Hope it helps?