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, using an electric mixer, cream the 1 cup softened butter and granulated sugar together 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.
Add 3 tablespoons milk and mix until the dough comes together and is smooth.
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 hand.
Bake for 8 to 10 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 before transferring 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 1 teaspoon vanilla extract and 2 tablespoons milk to the frosting and beat until fluffy. Add more milk if needed to reach desired consistency.
Once the cookies are completely cooled, spread or pipe the vanilla frosting on top of each cookie.