Preheat the oven to 350 degrees F (175 degrees C).
Cream butter and brown sugar for cookies together in a bowl with an electric mixer. Stir in pumpkin, then egg. Add baking soda, baking powder, vanilla extract, salt, cinnamon, and nutmeg. Stir in flour. Mix in walnuts and raisins.
Drop spoonfuls of dough 2 inches apart onto ungreased cookie sheets.
Bake each sheet on the bottom rack in the preheated oven for 6 to 7 minutes; continue to bake on the top rack until edges are golden, 6 to 7 minutes more. Continue with remaining cookie sheets in this manner. Let cookies cool on the cookie sheets for 5 minutes, then transfer to a cooling rack to cool completely.
While the cookies cool, melt brown sugar, milk, and butter for frosting in a medium pot over low heat. Bring to a boil; continue to boil mixture for 2 minutes, stirring constantly. Remove from heat and cool for 3 to 10 minutes. The longer the mixture is cooled, the thicker the frosting will be on the cookies.
Add powdered sugar and vanilla extract to the frosting mixture, stir well, and drizzle or spread onto cooled cookies. Allow the cookies to cool for 1 to 2 hours before stacking.