sam2- you are absolutely correct. Nowhere does the Shulcan Aruch- much less the Mishnei berurah- say anything about “two covers’ on the head.(Siman 2 and siman 91) I truly don’t know where this idea comes from.
chilled- the mishne berurah brings down the words of the chaya odo but he is talking about ONE cover, not two. (that cover, they say, shoulds be a full hat, not a small hat, as you would walk in the streets)