Preheat the oven to 325°F (163°C). Line the bottoms of three 9-inch cake pans with parchment paper. Grease the sides with cooking spray.

Sift together flours, cocoa powder, baking soda, and salt in a medium bowl. Set aside.

Microwave buttermilk just until warm, about 30 seconds. Stir in vanilla. Set aside.

Beat together brown sugar and butter in a large bowl with an electric mixer until light and fluffy, about 5 minutes. Add eggs one at a time, mixing thoroughly after each addition. Blend in flour mixture and buttermilk in 3 additions until fully combined. Let the mixer run on low speed for 5 minutes, then divide batter evenly between the prepared cake pans.

Bake in the preheated oven until a toothpick inserted into the center of cakes comes out clean, 30 to 35 minutes. Cool cakes in the pans for 15 minutes, then invert onto wire racks to cool completely, at least 30 minutes. Cover cakes with plastic wrap and chill in the refrigerator, 1 hour to overnight.

Meanwhile, make frosting: Combine sugar, milk, flour, and salt in a medium saucepan. Cook over medium heat, whisking constantly, until mixture thickens and begins to simmer, about 10 minutes. Pour into a medium bowl and chill in the refrigerator for about 45 minutes, stirring occasionally.

Beat butter in a large bowl with an electric mixer until smooth, about 5 minutes. Slowly add chilled milk-flour mixture, 1/4 cup at a time, beating well after each addition and scraping down the sides of the bowl. Mix in vanilla.

Level cake layers evenly using a serrated knife. Spread about 2/3 cup frosting onto a cake layer; repeat with second and third cake layers, stacking layers as you frost. Cover the outside of cake with a thin coat of frosting to catch any crumbs, then place in the refrigerator to chill for about 20 minutes.