Preheat oven to 350 degrees. Lightly butter and flour a 9-by-13-inch baking pan. In a large bowl, using an electric mixer, beat 1 1/2 sticks (3/4 cup) butter and sugar on high until light and fluffy, about 5 minutes. Beat in eggs, one at a time, until combined.
In a large bowl, whisk together flour, baking powder, baking soda, and 3/4 teaspoon salt. With mixer on low, add flour mixture to butter mixture in three additions, alternating with two additions buttermilk, and beat to combine, scraping down bowl as needed. Beat in vanilla. Transfer batter to pan and smooth top.
Bake until cake is golden, puffed, and a toothpick inserted in center comes out clean, 35 minutes, rotating pan halfway through. Let cool completely in pan on a wire rack.
Meanwhile, in a medium bowl, whisk together cocoa and malted milk powders. Add boiling water, whisk until smooth, and let cool.
Using mixer, beat 2 sticks (1 cup) plus 2 tablespoons butter on high until smooth. Add cocoa mixture; beat until combined. Add chocolate and pinch of salt. With mixer on medium-high, beat until frosting is smooth and peaks form, 3 minutes. Frost cake; top with sprinkles if desired.
A completely cool cake is essential so that the frosting doesn't melt on contact. You can bake the cake a day ahead to ensure a fluffy frosting layer.