For the cookies: Preheat oven to 375 degrees and grease baking sheets. In a medium mixing bowl stir together flour, baking soda, and salt, set aside.
In a large mixing bowl with an electric mixer on medium speed cream butter and sugar until fluffy. Beat in egg and vanilla. Gradually add flour mixture alternating with buttermilk and mix just until blended.
Drop dough by rounded tablespoonfuls 2 inches apart onto prepared baking sheets. Bake for 10 to 12 minutes. Transfer cookies to cooling racks to cool completley.
For the icing: In a small mixing bowl on medium speed cream butter until smooth. On low gradually beat in powdered sugar, alternating with milk, adding more if needed. Add vanilla and beat until smooth and creamy.
Spread icing on cooled cookies and sprinkle with walnuts.