For the cookies:
4½ cups all-purpose flour
4½ tsp. baking powder
¾ tsp. salt
1½ cups (3 sticks) butter, at room temperature
1½ cups sugar
3 large eggs
5 tsp. vanilla extract
For the frosting:
5 cups confectioners’ sugar, sifted
1/3 cup (5 1/3 tbsp.) unsalted butter, melted
1 tbsp. vanilla extract
7-8 tbsp. milk (plus more, as needed)
Food coloring (optional)
To make the cookies, preheat the oven to 350˚ F. Line baking sheets with parchment paper or silicone baking mats. In a medium bowl combine the flour, baking powder and salt, and whisk together to blend. In the bowl of an electric mixer, combine the butter and sugar and beat together on medium-high speed until soft and fluffy, about 2-3 minutes. Beat in the eggs one at a time, mixing well after each addition and scraping down the bowl as needed. Blend in the vanilla. With the mixer on low speed, add in the dry ingredients mixing just until incorporated and evenly mixed. Cover and chill the dough for 1 hour.
When you are ready to bake the cookies, scoop a scant quarter cup of dough and roll into a ball. Flatten the ball slightly and place on the prepared baking sheet. Repeat with the remaining dough, spacing the cookies at least 2-3 inches apart. Bake about 10-12 minutes or just until set. The edges should be no more than very lightly browned, if at all. Let cool on the baking sheet for several minutes. Transfer to a wire rack to cool completely.
To frost the cookies, place the confectioners’ sugar in a medium bowl. Add the melted butter, vanilla, and milk to the bowl and whisk until smooth. Whisk in additional milk as necessary, 1 teaspoon at a time, until you reach your desired consistency. Tint with food coloring if desired. Use an offset spatula or spoon to frost the cooled cookies. Top with sprinkles if desired. Store in an airtight container.