cover crops for chicken paddock/ veggie bed

I have a 700sq yd former garlic plot, that is overgrown with grass and weeds.  I'm thinking I'll rotate my chickens through, first to graze out the grass, and then to till in the cover crop (which I'll plant after they've grazed the grass out).  I have approximately 200 days from today til my garlic goes in the ground (I'm not great at math but I realize I'm not going to get through a whole rotation this season, even if I increase my flock numbers).  

Looking for advice from folks who've used cover crops like this...   How did you make it work?  What crops did you/ your chickens prefer?  Any pitfalls I should watch out for?

Thanks for any and all advice.

My laying hens seem to crave brasicas and radishes things that deliver more sulfur for the yolk. If I fork out any perennial grass roots after a pass and then water the annual weed seeds com up and are their first choice which are removed very quickly so you cane move through more quickly. I am on my second round now but I am in climate zone 7b almost 8 so my growing season starts earlier. If it is an area that will not be planted I feed the hens bird seed and they bury enough in spring to feed them in fall  Then I feed them soaked wheat which grows all winter here and ripens in summer. If winter wheat is planted in your area you might try that after the garlic is planted to open new ground.