I'm curious, how does a stutterer want to be treated? Should one just stare and wait?
Guess so, just share the awkward moment with them while they pull funny faces trying to get a sound out, and carry on the conversation like normal.
The more you act like it's not a big deal the more it will not be a big deal.
Pretend that their funny word choices are normal, ie: I use synonyms to avoid words i know i'll struggle with. Will sometimes even phrase something like Yoda, or put another sound in front of a word, like "um" or a random vowel.
Laughing about it sometimes breaks the ice, puts the stammerer at ease and makes speech easier for them, but only do that with friends you know who won't be offended, people can be very sensitive about it.
If they want to say something and then say "never mind", never mind then.
The bane of my existence is a situation where you go in a circle and have to say something like in a meeting, the tension and buildup towards your turn is fucking horrible, and I WILL stutter worse than normal when it's my turn, from personal experience my suggestion would be to start with the stutterer if you absolutely have to do something like that, otherwise fuck clockwise introductions in meetings where people already know each-other.
Don't fucking mock them though, afrikaans teacher mocked me once in std 5. Fucking asshole.