Melt the 1/2 stick of butter in a saucepan. Once it starts sizzling, add the dry milk powder and toast, stirring, until it is really dark brown (darker than you think looks right) – this builds big brown-butter flavor without browning all the butter.
Add the heavy cream to stop the cooking so the milk solids stay perfectly browned. Let this brown-butter mixture cool to room temperature.
In a stand mixer, beat the 2 sticks of room-temperature butter with the salt and vanilla on high until smooth and creamy, about 1 minute.
Add half of the sifted powdered sugar, pulse first so it doesn't fly everywhere, then beat on high about 1 minute.
Add the room-temperature brown-butter mixture and the remaining powdered sugar. Pulse to combine, then beat on high until you have a luxurious, plush buttercream.
Dollop and spread generously over a chocolate sheet cake, pushing it into the corners. Top with sprinkles – basically the perfect birthday cake.
