In a large mixing bowl, cream butter and confectioners sugar. Add vanilla. Gradually add flour, mix well.
With lightly floured hands, shape Tablespoonfuls into 2 inch logs. Place 2 inches apart on ungreased baking sheets. Bake at 350 degrees for 9 to 11 minutes or until edges and bottom are lightly browned. Cool for 2 to 3 minutes before removing to wire racks.
In a microwave safe bowl, melt chocolate chips and shortening, stir until smooth. Drizzle chocolate over half of the cookies. Dip one end of remaining cookies into chocolate, then sprinkle with walnuts.
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