Preheat the oven to 325°F (163°C). Line a baking sheet with parchment paper.
In a large bowl, cream together the softened unsalted butter and granulated sugar until light and fluffy, about 3-4 minutes.
Add 1 teaspoon vanilla extract and mix until combined.
In a separate bowl, whisk together the all-purpose flour and salt.
Gradually add the flour mixture to the butter mixture, mixing until the dough comes together and is smooth.
Roll the dough into a log about 2 inches in diameter, wrap in plastic wrap, and refrigerate for 30 minutes to firm up.
Remove the dough from the refrigerator and slice into 1/4-inch thick rounds. Place the rounds on the prepared baking sheet about 1 inch apart.
Bake for 20-25 minutes or until the edges are lightly golden.
Remove from the oven and allow the cookies to cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
While the cookies cool, prepare the icing by whisking together the powdered sugar, milk, and 1/2 teaspoon vanilla extract in a small bowl until smooth.
Once the cookies are completely cool, spread or drizzle the icing over the tops of the cookies.
Allow the icing to set for at least 30 minutes before serving or storing.