To begin baking the cake, preheat the oven to a temperature of 325 degrees F (163 degrees C). In the meantime, prepare the cake pans by lining the bottom of three 9-inch cake pans with parchment paper and spraying the sides with cooking spray.
Next, prepare the cake batter by sifting together all-purpose flour, cake flour, cocoa powder, baking soda, and salt in a medium-sized bowl. Heat the buttermilk in the microwave until it is warm, which should take about 30 seconds, and then stir in the vanilla extract.
In a separate bowl, beat the brown sugar and butter together until it becomes light and fluffy, which should take approximately 5 minutes. Add eggs one at a time, ensuring that each addition is mixed thoroughly before proceeding to the next one. Then, blend in the flour mixture and buttermilk in three separate additions until fully combined. Allow the mixer to run on low for 5 minutes, and then distribute the cake batter between the prepared cake pans.
Place the cake pans in the preheated oven and bake until a toothpick inserted into the center of the cakes comes out clean, which should take 30 to 35 minutes. Once baked, let the cakes cool in the pans for approximately 15 minutes before turning them out and allowing them to cool completely on a wire rack for at least 30 minutes. Cover the cakes with plastic wrap and chill them for at least 1 hour or overnight.
While the cakes are cooling, prepare the frosting by whisking sugar, milk, flour, and salt together in a medium-sized saucepan. Cook the mixture over medium heat, whisking constantly, until it thickens and begins to simmer, which should take approximately 10 minutes. Pour the mixture into a medium-sized bowl and chill for about 45 minutes, stirring occasionally.
Afterward, beat the butter until it becomes smooth, which should take about 5 minutes. Slowly add the milk and flour mixture, 1/4 cup at a time, beating between each addition and scraping the sides of the bowl occasionally. Mix in the vanilla extract.
Level the cake layers before assembling the cake. Spread approximately 2/3 cup of frosting onto the first layer and smooth it into an even layer. Repeat the process with the second and third cake layers. Frost the outsides of the cake with a thin layer of frosting to catch any crumbs and chill for 20 minutes.
Finally, finish frosting the cake with the remaining frosting and then cut and serve.
