Heat the oven to 350°F and lightly butter a 5x7-inch baking dish.
Whisk together the flour, sugar, baking soda, and salt in a medium bowl.
In a 1-quart saucepan, melt 4 tablespoons of butter over medium-low heat, then stir in the water and cocoa powder until the mixture reaches a light simmer.
Pour the warm cocoa mixture over the dry ingredients and stir until no dry streaks remain.
Stir in the egg white, 2 tablespoons of milk, the vanilla, and ¼ teaspoon of cinnamon until the batter is smooth.
Pour the batter into the prepared dish and bake for 25 to 30 minutes, until a toothpick inserted in the center comes out clean.
Make the frosting
Melt 2 tablespoons of butter in a clean 1-quart saucepan over low heat.
Stir in the cocoa powder and 1½ tablespoons of milk, and keep stirring until the mixture just begins to simmer.
Take the pan off the heat and whisk in the powdered sugar, vanilla, and ⅛ teaspoon of cinnamon until smooth.
Stir in the chopped pecans.
Pour the warm frosting over the warm cake and spread it to the edges.
Let the cake cool until the frosting sets into a fudgy layer, then slice and serve.
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Notes
Start the frosting while the cake finishes baking. Have it ready when the cake comes out, so you can pour it while both are still warm. That's what lets it sink in and set into a fudgy layer.Spread it fast. The frosting sets as it cools, so pour it over the warm cake and spread it to the edges right away.Check the cake at 25 minutes. A toothpick in the center should come out clean or with a few moist crumbs.Expect a thin cake. A Texas sheet cake bakes up thin and dense, not tall.Stick with a 5x7 or 6x6-inch dish. Both give the same thickness and bake time. If you use a bigger dish, the batter spreads too thin and bakes faster.If doubling this recipe, bake in an 8x8 inch pan.