In a large bowl, cream butter and sugar until light and fluffy. Beat in egg and vanilla. Combine the flour, baking powder and salt; gradually add to creamed mixture and mix well. Stir in walnuts. Cover and refrigerate for 1 hour or until easy to handle.
On a floured surface, roll out to 1/8-in. thickness. Cut with floured 2-in. cookie cutters. Place 1 in. apart on ungreased baking sheets. Bake at 375° for 6-8 minutes or until edges are golden brown. Remove to wire racks to cool completely.
In a small bowl, combine confectioners' sugar and milk; stir until smooth. If desired, tint with food coloring. Spread and pipe glaze over cookies; place on waxed paper. Let stand until set.
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