For the cookies:
4½ cups all-purpose flour
4½ teaspoons baking powder
¾ teaspoon salt
1½ cups (3 sticks) butter, at room temperature
1½ cups sugar
3 large eggs
5 teaspoons vanilla extract
For the frosting:
5 cups confectioners’ sugar, sifted
⅓ cup (5 tablespoons plus 1 teaspoon) unsalted butter, melted
1 tablespoon vanilla extract
7-8 tablespoons milk (plus more, as needed)
Food coloring and sprinkles, optional
Line baking sheets with parchment paper or silicone baking mats and set aside. Whisk together flour, baking powder and salt in a medium bowl.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and sugar together on medium-high speed until light and fluffy, 2-3 minutes. Beat in the eggs one at a time, mixing well after each addition. Blend in the vanilla and reduce mixer speed to low. Gradually add in the flour mixture until just incorporated. Cover and chill the dough for one hour.
Once the dough has been chilled, preheat oven to 350º. Scoop a scant quarter cup and roll into a ball. Flatten into a thick disk and place on the prepared baking sheet. Repeat with remaining dough, spacing the cookies at least 2 inches apart.
Bake for 10-12 minutes until just set, taking care not to overbake. The edges will be barely browned, if at all. Allow to cool on the baking sheet for several minutes before transferring to a wire rack to cool completely.
To make the frosting, place the confectioners’ sugar in a medium bowl. Whisk in the melted butter, vanilla and milk. Continue whisking until smooth, adding more milk as needed, a tablespoon at a time, to reach your desired consistency. Tint with food coloring, if desired.
Use an offset spatula to frost the cookies, and top with sprinkles if desired. If the frosting becomes too thick to be workable, whisk in additional milk. Store in an airtight container.