Place balls on greased cookie sheets, and press center of cookie with thumb.
Bake at 350 degrees F (175 degrees C) for 8 minutes. Press center of cookie again, bake until lightly browned 4 to 6 minutes. Cool cookies on racks.
To make frosting, begin by blending the water, shortening, vanilla, and 1/4 cup softened butter or margarine together thoroughly. Slowly add confectioners' sugar, and mix until smooth and creamy. More water may be needed to make desired consistency.