Preheat oven to 350°F. Line 2 cookie sheets with parchment paper.
Roll dough into 1-inch balls and press to fatten into oval shapes. Pinch ends with your fingers and pat into a football shape about ¼ inch thick. Bake for 9 to 11 minutes, or until edges are golden. Transfer cookies to rack to cool completely before frosting.
In a bowl, with an electric mixer, combine frosting and 1 cup powdered sugar and beat until smooth. Spread each cookie with a heaping teaspoon of frosting. Let stand for icing to set, about 30 minutes.
In a small bowl, combine remaining 1 cup powdered sugar and vanilla. Stir in water, 1 teaspoon at a time, until icing is thick and smooth. Transfer to a pastry bag fitted with a small round tip and squeeze a line of icing down the middle of cookies. Cross the line with 4 short perpendicular lines to resemble the laces on a football.