Preheat the oven to 350 degrees F (175 degrees C). Lightly grease cookie sheets.

Beat 1 cup softened butter and 1/2 cup sugar together in a large bowl with an electric mixer until smooth. Add egg yolks, and beat well. Add flour and mix until combined.

Roll dough into walnut-size balls; roll balls in egg white, and then in the chopped nuts. Place cookies 2 inches apart onto prepared cookie sheets; press center of cookie with thumb.

Bake cookies in the preheated oven for 8 minutes; press center of cookie again and continue to bake until lightly browned, 4 to 6 minutes. Cool cookies on racks.

To make the frosting: Blend water, shortening, vanilla, and 1/4 cup softened butter together; slowly add confectioners' sugar and mix until smooth and creamy. More water may be needed to make desired consistency.