You get an unfair email from the customer addressed to you. You publicly hear an opinion about you which you definitely don’t agree with. You face negative feedback which has to do much more with the “negative” part than with the “feedback” one. You are accused of something you don’t feel responsible for. I guess everyone was in those kind of situations. And it hasn’t happened only once.
Then you want to express your whole indignation. You want to answer immediately.
Calm down. Don’t let emotions to play first fiddle. If you can talk face to face about the situation wait until you can discuss merits, not emotions. If you can’t talk don’t rush to reply the email instantly. Wait. And don’t do the whole thing publicly.
Other way you won’t get what the other persons wanted to tell you. You won’t move closer to a problem solution but rather keep it where no consensus can be achieved and no one can learn anything.
Calm down, unless you keep your emotions boiling on purpose, which should be very rare situation by the way.