Preheat the oven to 375°F (190°C). Line two baking sheets with parchment paper.
In a medium bowl, whisk together the flour, baking soda, baking powder, and salt. Set aside.
In a large bowl, using an electric mixer, cream together 1 cup softened butter and 1 1/2 cups granulated sugar until light and fluffy, about 3 minutes.
Beat in the egg and 1 teaspoon vanilla extract until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined.
Add 3 tablespoons of milk and mix until the dough comes together. If the dough is too dry, add the remaining tablespoon of milk.
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.
Bake for 8 to 10 minutes or until the edges are just beginning to turn golden. Remove from oven and let cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
To make the frosting, beat together 1/4 cup softened butter and 2 cups powdered sugar until smooth and creamy.
Add 2 tablespoons milk and 1 teaspoon vanilla extract and beat until fluffy. If the frosting is too thick, add a little more milk, 1 teaspoon at a time, until desired consistency is reached.
Once cookies are completely cooled, spread the vanilla frosting evenly over each cookie using a knife or spatula.