Preheat the oven to 375°F (190°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, cream the 1 cup softened butter and granulated sugar together using an electric mixer on medium speed until light and fluffy, about 3-4 minutes.
Beat in the egg and 1 teaspoon vanilla extract until well combined.
Gradually add the dry ingredients to the wet ingredients, alternating with the 1/4 cup whole milk, beginning and ending with the dry ingredients. Mix until just combined.
Roll the dough into 1-inch balls and place them 2 inches apart on the prepared baking sheets. Flatten each ball slightly with the bottom of a glass or your palm.
Bake for 8-10 minutes, or until the edges are lightly golden but the centers are still soft.
Remove from the oven and let the cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely.
To make the frosting, beat together the 1/4 cup softened butter and powdered sugar in a medium bowl until smooth and creamy.
Add the 2 tablespoons whole milk and 1 teaspoon vanilla extract, and beat until the frosting is light and fluffy.
Once the cookies are completely cool, spread the vanilla frosting evenly over the tops using a knife or spatula.
Allow the frosting to set for 15 minutes before serving.