Preheat the oven to 350 degrees F. Line a 9x13 baking pan with parchment paper and set aside.
In a large bowl, whisk together flour, cocoa, baking soda, baking powder, salt and sugar.
Add the melted butter, milk, eggs and water. Mix until the wet ingredients are fully incorporated into the dry ingredients.
Transfer the batter to the pan and spread evenly. Bake for 28 to 32 minutes, until a toothpick inserted in the center comes out with little to no crumbs. Cool completely on a wire rack before frosting.
In the bowl of a stand mixer, beat the butter on medium speed until pale and fluffy, about 3 minutes. Add powdered sugar, cocoa and salt and mix on medium-low speed until well combined. Add vanilla and milk, mixing just until combined, scraping down the bowl as necessary.
Transfer the frosting to the cooled cake and spread in an even layer over the top. Garnish with chocolate sprinkles and serve.