Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking soda, baking powder, and salt. Set aside.
In a large bowl, beat the softened butter and granulated sugar together with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Add the egg and 1 teaspoon vanilla extract to the butter mixture and beat until combined.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined.
Mix in the sour cream until the dough is smooth.
Using a tablespoon or cookie scoop, drop rounded balls of dough onto the prepared baking sheets, spacing them about 2 inches apart.
Bake in the preheated oven for 8 to 10 minutes, or until the edges are lightly golden but the centers are still soft.
Remove the cookies from the oven and allow them to cool on the baking sheets for 5 minutes before transferring to a wire rack to cool completely.
To make the frosting, beat the softened butter in a medium bowl until creamy.
Gradually add the powdered sugar, mixing on low speed until combined.
Add the whole milk and 1 teaspoon vanilla extract and beat on high speed until the frosting is smooth and fluffy.
Once the cookies are completely cooled, spread the vanilla frosting evenly over the tops of the cookies using a knife or spatula.