Preheat the oven to 325°F (163°C). Line two baking sheets with parchment paper.
In a large bowl, cream the softened butter and granulated sugar together until light and fluffy, about 3-5 minutes.
Add the salt and gradually mix in the all-purpose flour until the dough comes together and is smooth.
Roll the dough into a log about 2 inches in diameter, wrap in plastic wrap, and chill in the refrigerator for 15 minutes.
Remove the dough from the refrigerator and slice into 1/4-inch thick rounds. Place the rounds on the prepared baking sheets about 1 inch apart.
Bake the cookies for 20-25 minutes or until the edges are lightly golden. Remove from the oven and let cool completely on a wire rack.
While the cookies cool, prepare the icing by whisking together the powdered sugar, milk, and vanilla extract in a small bowl until smooth and pourable.
Once the cookies are completely cooled, drizzle or spread the icing over the top of each cookie. Allow the icing to set before serving.