For a frosting to dry you have to reduce the amount of fat way down. Although I don't have an exact recipe for you there was one that my grandma used on molasses cookies and as I remember the base was egg whites whipped, add flavor and sifted powder sugar. Depending on the humidity it would dry quite quickly so we had to frost them fast. Royal icing is another one with just milk flavor and sifted powder sugar.
I like to use a glaze on my spritz cookies that is a small amount of melted butter(if you want to keep it white use crisco) flavor, hot water and sifted powder sugar. This one drys at it cools so you must keep it over hot water, like a double boiler. You can pour it over(and that means put the cookies on a cooling rack with wax paper under or brush it on with a new nylon brisle paint brush. If you need an exact recipe I will try and find one.