Preheat the oven to 350°F (175°C) and grease a 10×15-inch jelly roll pan.
Melt the butter in a saucepan over medium heat and continue cooking until golden brown with a nutty aroma.
Remove from heat and whisk in the cocoa powder and water.
Bring the mixture to a gentle boil, then remove from heat.
In a large bowl, whisk together flour, sugar, baking soda, and salt.
Pour the warm brown butter mixture over the dry ingredients and stir until combined.
Add the eggs, sour cream, and vanilla and mix until smooth.
Pour the batter evenly into the prepared pan.
Bake for 18–22 minutes, or until a toothpick inserted into the center comes out clean.
While the cake bakes, melt the butter for the frosting in a saucepan.
Whisk in cocoa powder and milk and bring the mixture to a gentle simmer.
Remove from heat and stir in vanilla and powdered sugar until smooth.
Fold in the chopped pecans.
Spread the warm frosting over the cake as soon as it comes out of the oven.
Let the cake cool before slicing into squares and serving.