To make the cookies, preheat the oven to 375°F (190°C). Line a baking sheet with parchment paper.

In the bowl of a stand mixer fitted with the whisk attachment, whisk together the flour, granulated sugar, cocoa powder, baking soda, baking powder, and salt. Fit the mixer with the paddle attachment, add the butter, and beat on low speed until incorporated, about 2 minutes, then beat in the egg and egg yolk. Raise the speed to medium and beat until the dough comes together, about 2 minutes.

Turn the dough out onto a well-floured surface and roll out 1/4 inch (6 mm) thick. Using a 2-inch (5-cm) round cutter, cut out cookies. Transfer the cookies to the prepared baking sheet, spacing them about 2 inches (5 cm) apart. Gather up the scraps of dough, reroll, and cut out more cookies, refrigerating the dough for 15 minutes if it gets too warm.

Bake until the cookies are firm to the touch, 8–10 minutes. Transfer the baking sheet to a wire rack and let the cookies cool on the sheet for 5 minutes, then transfer the cookies to the rack and let cool completely.

Meanwhile, make the filling: In the clean bowl of the stand mixer fitted with the clean paddle attachment, beat together the butter, confectioners’ sugar, milk, and vanilla on medium-high speed until smooth and well combined, about 3 minutes. Transfer to a pastry bag fitted with a 1/2-inch (12-mm) tip.

Pipe a layer of the filling (about 2 teaspoons) on the flat side of half of the cookies. Top with the remaining cookies, flat side down, and gently press together.