Combine the starter, milk, and flour in a mixing bowl. Cover and let rest at room temperature for 2 to 3 hours. It should expand slightly, but won't appear bubbly or active at this point.
Preheat the oven to 350°F. Grease an 8 inch square pan. If using a glass pan, preheat to 325 degrees F.
In a mixing bowl, beat together the sugar, oil, vanilla, salt, baking soda, cocoa powder and espresso powder. Add the egg and beat until well blended. You can do all the beating with a spoon or scraper – no electric mixer necessary.
Scrape the starter mixture into the chocolate mixture and stir until well-blended.
Pour the batter into the prepared pan and bake for 30 minutes (40 minutes if using 325) or until cake springs back lightly when touched and a skewer inserted in the center comes out clean.
Frosting: Mix the butter, sour cream, sugar, cocoa, salt and 2 tablespoons of the half and half until it all starts to come together as a thick sludge.
When everything is sticking together, add the vanilla. Increase mixer speed and beat until smooth, scraping bowl and adding more half and half (or milk and cream) as needed until you get your desired texture consistency. Spread over the cake.