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I can share my experience with chickens- I originally moved their fencing around and tried to regrow my grass in areas for them to graze on. (I have a very large garden for them to run in.)
I also had a LOT of chickens at the time. I have downsized, but what I discovered is that they eat the new grass so fast and scratch it up so quickly that eventually the grass is gone.
I then tried digging up sod one square per day or two and depositing it in the run like a pizza, for them to clean off, then returning the now-bare chunk of dirt to a hole. They loved it, but it was a lot of work.
Now I have found what works for me. I use yard scissors to clip (or pull up with my hands) 2-3 inch grass clippings, clover, dandeliion leaves, etc.
I give this to them every day. They get greens for health and I don't have to do much work. I have a tiller, too, but I don't want to kill all those earthworms for nothing because they will scratch it up too fast.
Rotational grazing might work for you.
If you divide your total area into paddocks (there was a thread on that recently), you can keep them off the land long enough for it to grow whatever crop you desire. It is also called the Balfour method.
You might plant clover, since I have noticed that my chickens go for the clover first, then eat the grass when that is gone. They also love dandelion leaves. I have grown millet for them and let them get the seeds off themselves when ready (when it was a garden and they were in a run).
Also- please be sure to note that you want to keep any matter you give them short, as long fibrous material (like long grass or hay) can cause impacted crop.