Heat oven to 350°F. Line a 9x13-inch pan with parchment and lightly grease.
Whisk flour, baking powder, baking soda, and salt in a bowl; set aside.
Cream butter and sugar until pale and fluffy, about 2 to 3 minutes.
Beat in eggs one at a time. Mix in vanilla and milk until smooth.
Add dry ingredients on low speed just until combined. Do not overmix.
Spread batter evenly in pan. Smooth the top with an offset spatula.
Bake 18 to 22 minutes, until edges are lightly golden and center is set.
Cool completely in the pan on a rack before frosting.
Beat frosting butter until creamy. Gradually add powdered sugar, vanilla, and salt.
Stream in cream until fluffy and spreadable, 2 to 3 minutes. Tint if desired.
Spread frosting over cooled bars. Add sprinkles. Chill 10 minutes, then slice.