Preheat oven to 375°. Cream butter and sugar until light and fluffy. Beat in egg and vanilla. In a separate bowl, whisk flour, baking soda and salt; add to creamed mixture alternately with buttermilk, beating well after each addition.
Drop by rounded tablespoonfuls 2 in. apart onto greased baking sheets. Bake until edges are lightly browned, 10-12 minutes. Remove to wire racks to cool.
For frosting, combine butter, confectioners' sugar, milk and vanilla; beat until smooth. Spread over cookies; if desired, sprinkle with chopped walnuts.