For chocolate cookies: Preheat oven to 375 degrees, set rack in middle of oven, and line a baking sheet with parchment paper. In the bowl of a stand mixer, combine flour, sugars, cocoa powders, baking soda, baking powder, and salt. Add butter and mix for 1 min. on low speed. Add egg and mix for 2 more min. Scoop dough to form balls (2 tsp. each) and place on baking sheet about 2 inches apart. Use the bottom of a glass to slightly flatten each ball. Bake 9 min., rotating sheet halfway through. Let cool for 10 min., then move to a wire rack. Repeat with remaining dough.
For creamy filling: In the bowl of a stand mixer, mix butter and shortening on low for 30 seconds, or until combined. Add powdered sugar 1/2 cup at a time, mixing after each addition. Add milk and vanilla, mix on low for 30 seconds, then beat on high speed for 2 to 3 min., or until filling is light and fluffy.
To assemble: Scoop frosting into a quart-sized zip top bag and cut off one corner. Lay out half of the cooled cookies, bottom-side up, and use bag to pipe about 1 Tbsp. of filling onto each. Sandwich the filling with remaining cookies and press together. Store leftover cookies at room temperature in an airtight container.