To make the cookies, cream the butter and sugar until light and fluffy. Beat in eggs and vanilla. In a separate bowl, stir together the flour, cocoa, baking soda, and salt. Add the dry ingredients to the wet and stir until combined.
Using a teaspoon of dough, roll the dough into balls. Chill them for 4 hours to overnight. Bake at 350 F for 7-8 minutes. Do not overbake. Remove from the oven and let the cookies cool on the cookie sheet for 2 minutes before removing to a wire rack to cool completely.
To make the frosting, beat the butter and pumpkin in a mixer on high for 2 minutes. Beat in pumpkin pie spice, vanilla and powdered sugar. Use milk or additional powdered sugar to thicken or thin the frosting as needed.
To assemble the cookies, spread a dollop of frosting on the bottom of one cookie and top with a second cookie.