Preheat your oven to 350°F and grease a 7x5-inch baking dish.
In a small bowl, whisk together flour, cocoa powder, baking powder, and salt.
In a medium bowl, cream butter and sugar until light and fluffy. Whisk in egg white and vanilla until well-combined.
One at a time, stir in half of the flour mixture, all of the milk, and the remaining flour mixture, mixing until just-combined after each. Stir in hot coffee (or hot water) and mix until combined. Mixture will be thin.
Pour into prepared baking dish. Bake for 14 to 18 minutes, or until a toothpick inserted into the center of the cake comes out clean. Cool for at least five minutes before starting the glaze.
In a 1-quart saucepan, combine butter, milk, corn syrup and vanilla. Heat over medium heat, whisking until butter is melted and ingredients are mixed. Add chocolate and whisk until chocolate is melted and incorporated. Turn heat down to low and add sifted powdered sugar. Whisk until sugar melts into the chocolate mixture. Remove from heat and pour immediately over the slightly cooled cake. Add sprinkles if desired.
Allow glaze 5 to 10 minutes to set before serving the cake in the baking container. Slice and enjoy with a strong cup of coffee.
Make sure to wait until the cake is done baking before starting the glaze. While it doesn’t completely harden, the glaze does set as it cools and will not be pourable after sitting for more than a couple of minutes.