Preheat the oven to 350°F (175°C). Butter three 9-inch cake pans, line the bottoms with parchment rounds, and butter the top of the parchment. Set the pans aside.
In a small bowl or liquid measuring cup, whisk ½ cup unsweetened cocoa powder with ½ cup lukewarm water until smooth.
In a separate small bowl or measuring cup, whisk ½ cup buttermilk with the remaining ½ cup lukewarm water until combined.
Sift 1 ½ cups cake flour and ¾ teaspoon baking soda together into a medium bowl; set aside.
In the bowl of a mixer, beat 1 stick (8 tablespoons) butter (room temperature), 1 cup sugar, and ½ cup packed brown sugar on medium-high speed until pale yellow and fluffy, about 5 minutes.
With the mixer running, add the 2 large eggs one at a time, beating well after each addition and scraping down the bowl as needed.
With the mixer on low, add the cocoa–water mixture and mix until just combined.
Add the flour mixture in three additions, alternating with the buttermilk mixture in two additions (begin and end with the flour). Mix on low and stop as soon as each addition is incorporated, scraping the bowl as necessary; do not overmix.
Divide the batter evenly among the three prepared pans (about 1 ½ cups batter per pan) and smooth the tops.
Bake the cakes for 16–18 minutes, rotating the pans halfway through baking, until the cakes spring back lightly when pressed or a toothpick inserted near the center comes out clean.
Cool the cakes in their pans on a wire rack for 10 minutes. Run a knife around the edges, invert each cake onto the rack, peel off the parchment, and let the layers cool completely on the racks.
Make the frosting: in a medium saucepan over medium-low heat, melt the remaining butter. Stir in the ½ cup packed brown sugar and ½ cup unsweetened cocoa powder (use the amounts from the ingredient list as written) until the sugar dissolves and the mixture becomes smooth and glossy. Continue stirring and cook until the mixture is hot and begins to bubble, then cook for 1 additional minute. Remove from heat.
Transfer the frosting to a bowl and chill in the refrigerator until thickened but still spreadable, about 1 to 1 1/2 hours, stirring occasionally to check consistency.
To assemble the cake, place one cooled layer right-side-up on your serving plate. Spread about ½ cup of the chilled frosting evenly to the edges. Place the second layer on top and spread another ~½ cup frosting evenly. Place the third layer on top and use the remaining frosting to cover the top and sides of the cake. If the frosting is too firm while assembling, let it sit at room temperature briefly and stir to soften.
Chill the assembled cake briefly to set the frosting if desired, then slice and serve.