Preheat the oven to 350 degrees F (175 degrees C). Grease cookie sheets.
In a large bowl, cream together shortening, 1/2-cup brown sugar, and white sugar. Mix in pumpkin, egg, and vanilla.
Sift together flour, baking soda, baking powder, cinnamon, and salt; mix into the creamed mixture. Stir in walnuts.
Drop dough by heaping spoonfuls onto the prepared baking sheets. Bake for 10 to 12 minutes in the preheated oven. Cool on wire racks.
In a small saucepan over medium heat, combine the 3 tablespoons butter and 1/2 cup brown sugar. Bring to a boil; cook and stir for 1 minute, or until slightly thickened. Cool slightly, then stir in the milk, and beat until smooth. Gradually stir in 2 cups confectioners' sugar until frosting has reached desired consistency. Spread on cooled cookies.