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