Place the chocolate and cocoa powder in a medium bowl. In a small saucepan over medium high heat, combine the sugar, butter and water. Heat, stirring occasionally, until the sugar and butter have melted. Pour the mixture over the chocolate and cocoa powder whisking until the chocolate is smooth. Then whisk the milk, egg, egg yolk and vanilla into the chocolate.
In a small bowl, combine the flour, baking powder, baking soda and salt. Stir the dry ingredients into the wet ingredients until fully combined. Let the batter rest at room temperature for 1 hour or up to 3 days in the refrigerator.
Preheat the oven to 350 degrees F. Line a 12-cup muffin pan with paper liners.
Divide the batter into the muffin cups and bake for 30 minutes until the cupcakes spring back when pressed in the center. Allow the cupcakes to cool completely before removing them from the pan.
For the frosting, combine the granulated sugar and egg whites in a heatproof bowl over a saucepan of simmering water over low heat. (Make sure the water is not touching the bottom of the bowl.) Heat the mixture until it is hot to the touch and the sugar has melted, about 3 to 4 minutes.
Transfer the sugar-egg white mixture to the bowl of a stand mixer fitted with the whisk attachment. Beat for 6-8 minutes until it becomes a light, white meringue. Mix in the butter a few chunks at a time until fully combined. Then add the confectioners’ sugar, salt, milk and vanilla extract and continue beating on medium speed until smooth and shiny.
Once the cupcakes have cooled, use a pastry bag fitted with a round or star tip to pipe the frosting or spread it with a spatula. Top with sprinkles after piping each cupcake.
The cupcakes can be stored up to 2 days at room temperature, but they taste best the day they are baked.