Preheat the oven to 375°F (190°C). Line 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 together the softened butter and granulated sugar until light and fluffy using an electric mixer, about 3 minutes.
Beat in the egg and 1 teaspoon vanilla extract until well combined.
Gradually add the dry flour mixture to the wet ingredients, mixing on low speed until just combined.
Add 2 tablespoons milk and mix until the dough comes together. If the dough is too sticky, add a little more flour, 1 tablespoon at a time.
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 in the preheated oven for 8 to 10 minutes, or until the edges are lightly golden but the centers remain 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.
While the cookies cool, prepare the vanilla icing by mixing the powdered sugar, 1 tablespoon softened butter, 1/2 teaspoon vanilla extract, and 2 teaspoons milk in a small bowl until smooth and spreadable. Add more milk if needed to reach desired consistency.
Once the cookies are completely cooled, spread the vanilla icing on top of each cookie and decorate with assorted colored sugar sprinkles as desired.
Allow the icing to set before serving or storing.