Well, to be honest - me too had to look up the word
"balmy". I don t know where you found that sentence but it is certainly not commonly used.
So, balmy means mild, healing and the noun "balm" means a medical creme.
I think "balmy September evening" means that it was a mild (with reference to the wheather) September evening.
But probably you shouldn t talk to your friends about meeting on a "balmy evening".
"Balmy" also has the sense of slow, relaxing, enjoyable to go along with the warm weather you get in September. It's not quite the summer but it still has the feeling of being summer.
In this case, the sentence implies that the person is referring fondly to an event in the past that was enjoyable.
"On the balmy September evening when we first visited, the tables outside (the restaurant) were all occupied."
"The outside tables were occupied on the balmy
September evening we first visited."
This sentence could be written more clearly. Icchan, you broke down the parts of the sentence perfectly. "...we first visited" is referring to the first time they went to the restaurant. I think it would be a much clearer sentence if they had written "when we first..."
As for balmy... balmy wiould probably be used more in writing than in speaking (although I'm sure there are some people who also use it in speaking). There are a lot of people who who do not know the meaning of the word balmy, and there are a lot of people who know that balmy has to do with weather but don't know exactly what kind of weahter balmy is.