Excellent ideas posted by all. If I were faced by such conditions, I'd look at perhaps building a couple of grow frames, perhaps each 4 x 8 and covered with hardware cloth. Alternate between frames, while one frame is resting, do a lasagna compost in it, and then sow mixed greens. Use a combination of grasses and high nutrient greens (kale and a lot of the other asian greens/cole crops, legumes, a few marigolds and calendula) I also suggest that you look into using deep litter in your run. That will give them healthy gut flora that can only enhance their digestion. The grow frames would allow you to concentrate the nutrient and moisture in a specific area. Perhaps you could also look at options such as a rain barrel, collecting dew with a catchement system. You can grow a lot of greens year round (both for your own use and to feed your birds) with a simple cold frame. I built a cattle panel hoop green house, and have that planted for my own use. Have been harvesting salads for the past 10 days for my own use. Expect that to produce for me through mid November, then ramp back into production in late Feb. Need to pick up some more lumber to complete a hay bale cold frame to plant more greens that will be providing late winter salads for my family. I am in growing zone 4, 45.6 latitude. Check out Eliot Coleman's excellent books about year round gardening.
If you want to give the flock full access to your grow frames for their digging pleasure, you could make some beds using hay bales to surround them. Then build up a lasagna bed inside the hay bale frame. Be sure to lightly mulch after you've seeded it. This will preserve moisture, and will also inoculate it with lots of weedy seeds. Keep it moistened and covered until the growth is well established. The hay bales will keep the chickens from scratching all of your compost out of the bed.