Heat your oven to 350 degrees F and line two baking sheets with parchment. Grate the zucchini and squeeze it tightly in a clean kitchen towel until very dry. Set aside.

Whisk flour, cocoa powder, baking soda, and salt together in a medium bowl. In a large bowl, beat the softened butter and granulated sugar on medium speed for about 2 minutes until creamy. Add the egg, vanilla, and milk. Mix until smooth.

Stir the flour mixture into the butter mixture until just combined, then fold in the squeezed zucchini. The dough will be soft but should hold its shape when scooped.

Use a 1 tablespoon cookie scoop to portion the dough. Space mounds about 2 inches apart on the lined baking sheets. Use the back of a spoon or your fingertips to gently flatten each mound slightly so the cookies bake up thin and flat. Flat cookies are much easier to sandwich.

Bake for 8 to 10 minutes, until the edges look set and the tops are no longer shiny. Do not overbake. They will feel a little soft in the center when you pull them, and they firm up completely on the rack. Cool on the pan for 4 minutes, then transfer to a wire rack to cool fully.

While the cookies cool, make the buttercream. Beat 1/2 cup softened butter on medium speed for 1 minute until smooth. Add the sifted powdered sugar one cup at a time, mixing on low so it does not cloud up. Add the vanilla and 1 tablespoon of cream or milk and beat on medium-high for about 2 minutes until fluffy and spreadable. Add a tiny bit more cream if it feels too stiff to pipe.

Match up the cooled cookies into pairs of similar size. Spread or pipe a generous layer of buttercream onto the flat side of one cookie in each pair, then press the flat side of the second cookie on top. Press gently until the filling just reaches the edges. Repeat with the remaining cookies and let the sandwiches sit for about 10 minutes before serving so the buttercream can settle.