Prepare cookies; Preheat oven to 375 degrees. Sift together flour, cocoa powder, baking soda, and salt into a medium bowl; set aside.
Place butter, shortening, and sugars into the bowl of an electric mixer fitted with the paddle attachment. Mix on high speed until smooth, about 3 minutes. Add egg; mix until pale and fluffy, about 2 minutes. Mix in half the flour mixture, then the milk and vanilla. Mix in remaining flour mixture.
Drop about 2 teaspoons dough onto baking sheets lined with parchment paper, spacing 2 inches apart. Bake until cookies spring back when lightly touched, 12 to 14 minutes. Transfer baking sheets to wire racks and let cool 10 minutes. Remove cookies from baking sheets and transfer to wire racks using a spatula; let cool completely.
Prepare filling; In the bowl of a mixer fitted with the paddle attachment, whip together cream cheese, butter and confectioners' sugar on medium speed until smooth, about 3 minutes. Add pumpkin, cinnamon, and nutmeg; whip until smooth, scraping down the bowl as necessary.
Pipe or spoon about 2 teaspoons filling on the flat sides of half the cookies. Sandwich with remaining cookies, keeping the flat sides down.