To make the donuts, preheat oven to 375 degrees.
In a medium mixing bowl, combine the flour, sugar, cocoa powder, baking soda, and espresso powder.
In a small bowl, beat together the vanilla, egg, sour cream, milk, and oil.
Stir the wet ingredients into the dry until just combined. Spoon in a greased donut pan.
Bake for 8 minutes or until the tops spring back when you touch them. Let the donuts cool in the pan before frosting.
To make the frosting, add sugar, butter, and milk to a small saucepan over medium heat and bring to a boil. Boil for exactly 60 seconds and remove from heat.
Stir in chocolate chips and continue stirring until the chocolate has melted. Use frosting immediately as it will harden quickly.
Dip the tops of the donuts into the frosting and remove to a cooling rack.
Immediately top with sprinkles.
Let sit for 10 minutes before serving.