Preheat oven to 350 degrees. Lightly grease a mini-donut pan.
In a medium-size mixing bowl, stir together flour, sugar, baking powder, and salt.
Stir in egg, milk, and melted butter until combined.
Transfer batter into a pastry bag fitted with a large circular tip. Pipe dough into donut molds, filling about 1/2 or 2/3 full.
Bake in the preheated oven for about 10 minutes.
Meanwhile, make the glaze by stirring together powdered sugar and cocoa powder in a small bowl. Add in water and stir until combined. Spread several tablespoons of sprinkles out on a small plate.
Remove from oven and remove donuts from pan. Dip the tops of the donut into the glaze and then into the sprinkles. Let cool completely on a wire rack.