Make the cookies: Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Whisk the flour, cocoa powder, baking powder, baking soda and salt in a medium bowl. Stir the milk, espresso powder and vanilla in a liquid measuring cup until dissolved. Beat the butter and granulated sugar in a large bowl with a mixer on medium-high speed until light and creamy, about 1 minute. Add the egg and beat until combined. Reduce the mixer speed to low; beat in half of the flour mixture, then the milk mixture, then the remaining flour mixture. Increase the mixer speed to medium and beat until smooth, about 1 minute.
Drop tablespoonfuls of dough about 2 inches apart on the prepared baking sheets (dip the spoon in water if the dough sticks); form into even mounds with damp fingers. Bake, switching the pans halfway through, until the cookies are set around the edges and the tops spring back when touched, about 10 minutes. Let cool 3 minutes on the baking sheets, then transfer to racks to cool completely.
Make the filling: Beat the mascarpone, cream cheese, 1/3 cup confectioners' sugar and the vanilla in a medium bowl with a mixer on medium speed until smooth and light, about 1 minute. Transfer half of the filling to a separate bowl. Stir the liqueur and espresso powder in a small bowl until dissolved, then add to the remaining filling and beat until combined. Transfer the fillings to separate large resealable plastic bags and snip a corner on each.
Flip over half of the cookies; pipe a circle of espresso filling around the edge of the cookies. Pipe the plain filling inside the circles and sandwich with the remaining cookies. Refrigerate until the filling is firm, about 1 hour. Dust with confectioners¿ sugar.