Prepare the oven and pan: Preheat the oven to 350°F (175°C). Lightly grease a 13 x 18-inch (33 x 46 cm) rimmed half-sheet pan, making sure to coat the corners.

Combine the dry ingredients: In a large mixing bowl, whisk together the flour, granulated sugar, and salt until evenly combined.

Heat the cocoa mixture: In a medium saucepan, combine the 1 cup (226 g) butter, water, and 1/4 cup (25 g) cocoa powder. Heat over medium heat, whisking occasionally, until the butter melts and the mixture comes just to a boil.

Add the hot mixture to the flour: Remove the saucepan from the heat and immediately pour the hot cocoa mixture over the flour mixture. Whisk until mostly smooth. The batter will be thick at first and then loosen as it is mixed.

Add the remaining wet ingredients: Whisk in the buttermilk, eggs, baking soda, and vanilla. Continue just until the batter is smooth and evenly chocolate-colored, with no dry flour pockets.

Fill the sheet pan: Pour the batter into the prepared pan and spread it evenly into the corners with a flexible spatula. The batter layer will be relatively shallow.

Bake the cake: Bake on the center rack for 17 to 20 minutes. The surface should spring back lightly when touched, and a toothpick inserted near the center should come out with a few moist crumbs but no wet batter.

Begin the frosting while the cake bakes: During the final 5 to 7 minutes of baking, place the 1/2 cup (113 g) butter, milk, and 1/4 cup (25 g) cocoa powder in a small saucepan over medium-low heat.

Heat the frosting base: Whisk until the butter melts and the mixture becomes smooth and steaming. Bring it just to a gentle simmer, then remove it from the heat.

Finish the frosting: Whisk in the powdered sugar, vanilla, and salt until smooth. Fold in the chopped pecans. The frosting should be glossy and pourable but thick enough to coat the back of a spoon.

Frost the warm cake: Remove the cake from the oven and place the pan on a wire rack. While the cake is still hot, immediately pour the warm frosting over the surface.

Spread quickly: Use a flexible spatula to guide the frosting into the corners. Work gently and quickly because the frosting begins setting as it cools.

Cool completely: Let the cake cool in the pan for at least 1 hour. The frosting should lose its wet sheen and feel set enough to slice without running.

Slice and serve: Cut the cooled cake into 24 rectangles or squares. Wipe the knife between cuts if needed for cleaner edges.