I keep seeing sources that list the leaves of bean plants as being safe for people to eat, and I haven't found anything that lists a chicken-specific reason to avoid them.
For the sake of clarity: I am talking about phaseolus vulgaris, the common green beans and string beans and kidney beans...
Maybe something like peas or beans? Both of those grow fast, and the greens are safe for chickens to eat if/when they poke into the run, or if the chickens get out. Neither one is likely to be invasive the way mint is. Peas like cooler weather, beans like warmer weather, so you might choose one...