1. Preheat the oven to 350 degrees. Coat an 8-inch square baking pan with cooking spray, then line the bottom and sides of the pan with parchment paper, leaving a 2-inch overhang on 2 opposite sides; set aside. Whisk together the ½ cup boiling water and cocoa in a small heatproof bowl; set aside. In a large bowl, or in a stand mixer fitted with the whisk attachment, whisk together the brown sugar, oil, and vanilla until well combined. Whisk in the egg, egg yolk, and buttermilk. Add the baking powder, baking soda, and salt; whisk to incorporate. Add the cocoa mixture (do not stir), then add the flour. Whisk vigorously until the batter is well combined, about 20 seconds.
2. Transfer the batter to the prepared baking pan. Bake until a wooden pick inserted into the center comes out with a few moist crumbs, about 25 minutes, rotating the pan halfway through the baking time. Let the cake cool completely in the pan on a wire rack, about 1 hour and 30 minutes. Lift the cake from the pan using the parchment paper overhang as handles; transfer to a platter. Discard the parchment paper, and set the cake aside.

Make the pistachio frosting:

1. Beat the butter, vanilla, and salt in a stand mixer fitted with the paddle attachment on medium speed until well combined, about 1 minute. Reduce the mixer speed to medium-low, and gradually beat in the powdered sugar, pistachio paste, and heavy cream, stopping to scrape down the sides of the bowl as needed. Increase the speed to medium-high, and beat until the frosting is light and fluffy, about 2 minutes. Spread the frosting evenly over the cake.