Beat butter and powdered sugar together on low for 30 seconds.
Increase speed to medium, and beat until well-combined, about 2-3 minutes.
Mix in chocolate and milk until completely incorporated.
Mix in vanilla.
Butter: I use salted butter in this recipe. If you use unsalted butter, I recommend adding a pinch or two of salt to the frosting.
When I soften my butter, I soften it to room temperature (about 72F) so that the butter is soft but not smooshy or overly soft.
Baking chocolate: I use unsweetened chocolate in this recipe. If you wanted a sweeter frosting, you could sub in bittersweet chocolate.
To melt the chocolate, I chop the chocolate and microwave it in a microwave-safe bowl for 30 seconds. I stir it, and then I microwave it in 15 second increments, stirring in between, until the chocolate has melted.
Milk: You can use milk (skim to whole), half-and-half, or heavy whipping cream. If using heavy whipping cream, you'll get a lighter, fluffier frosting. However, be careful not to overwhip it.
I recommend storing leftover frosting in the fridge.