In the bowl of a standing mixer fitted with the paddle attachment, beat butter and powdered sugar at a low speed until smooth, about 2 minutes.
Add vanilla and mix just until combined.
In another bowl, sift together flour and salt.
With the motor running, add milk to the whipped butter and about half of the flour. Beat until fully incorporated. Add the rest of the flour and mix just until a dough forms. If the dough is too sticky, add a spoon of flour. If it is too dry, add a spoon of water.
Divide the dough in half. Shape each half into a disk, wrap it with plastic food wrap. Refrigerate for 1 hour or until firm enough to roll.
Preheat the oven to 350°F/180°C.
Transfer the dough to a floured surface and roll to about a ½-inch/1-cm thickness (relatively thick so that they do not break).
Cut cookies into the desired shape and place spaced out on a baking sheet lined with parchment paper.
Bake for approximately 12-15 minutes, until golden at the bottom. Let cookies cool completely, preferably on a cooling rack.
Using a teaspoon or a piping bag, spread the filling on the bottom of half of the cookies and then cover it with the remaining cookies.
Garnish with powdered sugar and enjoy!