We move our chickens (+run!) regularly and grass recovers no problem.
If moving is not an option one solution I read was to give them a large area, but only 1/2 of it at a time, giving time for the other half to recover. Hoping to do this myself eventually, so would love to know if it works!
Else give up on grass and use a renewable surface eg bark, sharp sand that can be removed and replaced, composting the old stuff.
Sure others here will also have some useful help!