I think it's ridiculous when new people move into a home and immediately start complaining about what their neighbors are doing. Your chickens were there first!
I have an idiot neighbor kinda like this, and I love to do small, legal things that irritate the crap out of them. Such as releasing thousands of insects into my vegetable garden, then snickering when a pest control company shows up at their house a week later. Repeatedly.
I've also: parked my car in my lawn, stopped watering the front lawn, left my RV (which in all honesty I had planned to move) in my driveway for them to complain about, even though it's perfectly legal there. I put in a veggie garden in the front lawn, and had to tear out a bunch of tree roots to do that (no trees on my side of the fence). So, yeah. The neighbors leave me alone now, which is exactly the way I like it.
If they want to fabricate drama, I'll give them all the drama they want. In my quiet, passive aggressive and totally legal manner.
Good luck with your own neighbors.
I confess to doing similar.

My avatar is my bike on the front lawn.

Wanna gripe about me painting something for the back yard purple?
Ok now my bench on the front porch is purple with colorful butterflies stenciled on.
Gripe about hubs tow truck parked out front... Ok we got the law changed so even the really big one is legal there.
Don't like the deliberately chosen flowers I planted....ok now it is a sprinkled wildflower mess.
Yeah I don't like that guy.