Preheat oven to 350 degrees F (175 degrees C). Grease a 10x15-inch jelly roll pan.
Melt 1 cup butter in a saucepan over medium heat. Whisk in ⅓ cup cocoa powder, water, and espresso powder; mix until smooth.
Combine white sugar, brown sugar, flour, 1 ½ teaspoon cinnamon, and baking soda in a mixing bowl. Pour in the butter-cocoa mixture, vanilla extract, ½ cup milk, and eggs. Mix until fully incorporated. Pour the batter into the prepared baking sheet
Bake in the preheated oven until the cake springs back when touched lightly with a finger, about 18 minutes. Cool the cake in the pan on a wire rack.
To make the icing, melt ¼ cup butter in a saucepan or in the microwave. Transfer melted butter to a mixing bowl. Whisk in 3 tablespoons cocoa powder, coffee, ½ teaspoon cinnamon, and 1 tablespoon milk. Slowly mix in the sifted confectioners' sugar and beat until smooth; add milk as needed to reach the consistency you like. Spread the icing over the cooled cake. For easier serving, score the icing before cutting the cake: mark lines as a guide before you start to cut, and divide the cake into 32 pieces.
