In a medium bowl, beat the butter and sugar until light and fluffy. Stir in vanilla extract.

Add the flour and salt and beat until combined. Dough will be crumbly.

Wrap dough in plastic wrap and chill until firm, about 1 hour.

Preheat the oven to 325 F. Line two baking sheets with parchment paper.

Roll dough into balls about 1 ½ inches in diameter. Place balls onto the prepared baking sheets, about 2 inches apart. Gently press into the center of each ball with your thumb.

Bake cookies for 10 minutes. Remove from the oven and gently press in the centers with the end of a spatula or wooden spoon. Bake for another 10-15 minutes, until the edges are golden brown. Let cookies cool completely on the baking sheets before frosting.

In a small bowl, whisk together the powdered sugar and the milk, one tablespoon at a time. The icing should be smooth and spoonable, but not runny. Add more milk or powdered sugar as needed.

Once the cookies have cooled, use a spoon to scoop icing into the centers of each cookie. Let the cookies sit out overnight to allow the icing to firm up.