and while you may have moved away from the city for "peace and quiet", they may have moved away from the city to have their 6 hens...
yes, there are potential legal avenues to explore, mostly involving the law of "nuisance" which is ill defined at best, and very frustrating to those of us who own chickens - a way that a person's dislike can be used to force another to stop otherwise legal behavior on property the complainant doesn't own. Particularly when a new person buys into the area, then wants the pre-existing property owners to change the way they are using their own land to better fit the sensibilities of the new neighbor.
From that perspective, I'm sure you understand why that approach goes over like a lead balloon, typically.
Good fences and a neighborly conversation should help. A well managed chicken coop shouldn't "stink". Feel free to ask if the neighbor if they have spent any time on BYC - we'd be happy to help them find a management practice that's good for them, good for the birds, and good for odor control. If you can politely work it into the conversation.
/edit as to why they have hens??? Why does it matter? If anything, a hen serves a more objectively valuable purpose than a lizard, turtle, fish, hampster, dog, cat, or parakeet for most people. A hen produces something of value (admittedly, usually less valuable than the cost of maintaining the hen). Those other animals produce nothing of value, few keep cats for purposes of rodent control anymore, or dogs to protect their other animals. Nor is it for you to say that they should derive less subjective pleasure from their pet chickens than some other pet. Peopel are people, not everything is explainable. Cat People, Dog People, Coffee People, Tea People, there are even MORNING People! Scary, I know...