Preheat oven to 350 degrees. Grease 2 standard sized 6-ct. donut baking pans; set aside.
In a medium bowl, combine flour, cocoa powder, brown sugar, baking powder, baking soda, salt, and espresso powder. Whisk well, making sure there are no lumps left from the brown sugar.
Add egg, Coca-Cola, vanilla extract, apple cider vinegar, and melted butter. Whisk until well combined.
Spoon batter into a gallon-sized plastic zip-top bag and snip a small piece off one corner. Use the bag to pipe the batter into the donuts pans, filling each approximately 3/4 full.
Bake at 350 degrees for 10-12 minutes, until a toothpick inserted in the center comes out clean.
Allow donuts to cool in the pans for 5 minutes. Then remove to a wire cooling rack to cool completely.
For the glaze:
While the donuts are cooling, prepare the glaze by melting the chocolate chips and milk in a double boiler. Whisk until smooth.
Dip the tops of the cooled donuts in the glaze. Set on a wire cooling rack and allow the glaze to set up (if desired) or serve immediately.