Help! Melted butter frosting!

You have a few options here. You can:
Make another batch without milk and marry the two.
Beat in more confectioner's sugar until consistency is correct.
Chill it until hard and beat it until room temp.
Consider making it next time with sweetened condensed milk instead of milk.

I would strongly suggest avoiding the cornstarch/flour option. That's difficult territory because you'd be attempting a frosting out of "quick" pastry cream which is not exactly possible or delicious.

I don't know my suggestion would really work or not , but if you wish to give it a try take some cornstarch and mix it well in the frosting. have you heard about ermine frosting which uses AP flour or cornstarch, it might help bind the frosting rather than adding extra sugar and making it too sweet. It's upto you.

Actually, for the ermine frosting I cook cornstarch in milk on stove for few minutes to remove the rawness and then add other ingredients like butter, sugar, heavy cream and blend it well. In the hotline search option if you type ermine frosting , you can find out some perfect measurements. Good luck. Hope this helps.

I make buttercream all of the time. I always use room temp unsalted butter and I rarely use milk. When I do it's only about a teaspoon or so.

You are just going to have to add more powdered sugar until the consistency is right. I always get the best results with salted butter, that how my grandma made it so I am not going to argue with her. I know I have friends that use unsalted but...I might be tempted to add a pinch of salt if you used unsalted.