Use a stand mixer with a whisk attachment (and hand mixer works as well) and beat the butter on medium speed to lighten it a bit

In another bowl, combine the powdered sugar with the cocoa power using a whisk; if you have a lot of chunks then push it through a sieve to break it up

Add the powdered sugar/cocoa mixture into the butter, and carefully start to mix; once it starts to come together then you can raise the speed to medium

Slowly add the vanilla and some of the half and half so that it starts to come together and become smooth

Scrape down the sides of the bowl and then add the melted and cooled chocolate; mix together on medium speed to incorporate the chocolate well

Add more half and half of the frosting seems too thick; you are looking for a smooth, spreadable consistency