If your sourdough starter lives in the refrigerator, start prepping the day before to get it "happy" and bubbly. See notes in blog post for more info.
Stir together fed sourdough starter, milk, and flour in a medium bowl. Cover with plastic wrap and allow to sit at room temperature for 2-3 hours. It will probably not bubble, but will smooth out and may expand a bit.
Preheat oven to 350 degrees and grease a 9x13 baking pan.
In a large mixing bowl, combine the sugar, oil, vanilla, salt, baking soda, cocoa, and espresso powder. Mix on medium with your stand or hand mixer until well combined.
Beat in eggs, one at a time, allowing the first to become completely incorporated before adding the second.
Add the starter/flour/milk mixture to the bowl and combine, using the "stir" or low setting on your mixer. Gently combine the two mixtures until smooth and cohesive. (It will take a little while for the starter mixture to become fully blended with the chocolate mixture.)
Pour batter into the 9x13 pan.
Bake for 30-40 minutes, or until a toothpick inserted in the center comes out clean. The edges of the cake may be just starting to pull away from the pan.
Remove from oven, and set on cooling rack to begin cooling while you prepare the icing.
Sift confectioners' sugar into a large bowl.
Place a small sauce pan on your stove over medium heat, and add butter and yogurt. Heat until butter is melted.
Dissolve espresso powder in the hot water in a small bowl or mug, then add to sauce pan.
Bring mixture just to a boil, then immediately pour over confectioners' sugar in the large bowl.
Beat until smooth.
Pour icing over cake.
In a small microwave-safe bowl, place chocolate chips, 1 Tbsp milk, and light corn syrup. Microwave 10 seconds at a time, until chips soften. Stir until smooth.
Drizzle chocolate over coffee icing with a spoon, or use a plastic bag with the corner snipped off.
We found that chilling the cake in the refrigerator after adding the icing helped everything to set up nicely.