Cookies: Cream the butter, brown sugar and white sugar until fluffy. Beat in egg and vanilla. Stir in the pumpkin and mix well. Combine the dry ingredients and add to the creamed mixture and mix well. Stir in the walnuts. Drop by tablespoonfuls onto greased baking sheets. Bake at 350°F for about 8 – 10 minutes.
Frosting: Beat all the ingredients together until light and fluffy. Frost cooled cookies.