A person who can laugh at themselves is really hard to embarrass.

Embarrassment needs a gap between the image you are defending and what just happened. Someone who can name their own mess first has no gap left to expose, which is why the joke aimed at them never quite lands. It is not toughness, it is having less to protect. People also relax around it, because being near someone who cannot be caught out is restful.

Tell one story this week about a time you got something wrong, without softening the ending. Small is fine. Notice what happens in the room, and notice that the thing you were protecting turned out not to need protecting.