Preheat oven to 350°F (175°C). Line a 9"x 13"x 2" baking pan with parchment paper. Grease (or butter and flour) the paper and any exposed parts of the inside of the pan.*
Whisk together the flour, baking powder, baking soda, and salt. Set aside.
Using an electric mixer on medium speed, beat the butter and sugar until light and fluffy. Add the eggs, one at a time, mixing well after each addition. Mix in the vanilla.
Reduce mixer speed to low. Add the flour mixture in 3 portions, alternating with 2 portions of milk. Mix just until combined.
Transfer the batter to the prepared pan and spread evenly. Tap the pan on the counter a few times to get rid of large air pockets.
Bake 35 to 40 minutes, or until the cake is golden brown and a pick inserted into the center comes out clean.
Cool the cake completely in the pan on a wire rack.
To make the frosting:
Beat the confectioners’ sugar and butter until thoroughly combined.
Add 2 tablespoons of the milk and the vanilla. Continue beating, adding a small amount of milk at a time until the frosting is the desired consistency.
Spread the frosting on the cooled cake. Top with sprinkles.
Notes
*The parchment can be used to make a "sling" that you can use later to remove the cooled cake from the pan.
Store tightly covered in the refrigerator up to 5 days. Bring to room temperature for serving.