Preheat oven to 350°F. Grease 6 muffin cups or place 6 liners in a muffin pan. Set aside.
With a stand mixer or hand mixer, combine sugar, flour, cocoa powder, baking powder, baking soda, and salt. Add egg, milk, oil, and vanilla extract and beat for 2 minutes. Add hot coffee or hot water, and beat until just combined. Batter will be thin.
Divide evenly between 6 prepared muffin cups. Bake for approximately 15 minutes, until the tops spring back when you touch them, and a toothpick inserted into the center comes out clean. Transfer the pan to a wire rack to cool completely.
For the frosting:
Beat butter and powdered sugar for several minutes, until smooth. Add vanilla extract and milk, and beat for several more minutes, until light and fluffy. If the frosting seems a little too thick, beat in a splash of milk. If it seems too thin, add another tablespoon of powdered sugar.
Frost the cooled cupcakes using a decorating tip or a butter knife. Garnish with sprinkles, if desired.
The flavor of the coffee will not come through in the cupcakes. The coffee flavor will help intensify the chocolate flavor. If you'd rather not use coffee, you can substitute hot water.