In a large bowl add the unsalted butter, brown sugar, and white sugar. Beat until light and fluffy. Add in the egg, yolk, and vanilla. Beat until combined. Add the flour mixture 1/3 at a time until all of the flour is fully incorporated.
Prepare the thin glaze in a small bowl. Whisk together the powdered sugar, vanilla, and milk.Drizzle the icing over the cooled cookies. Allow the icing to set before serving.