If you can make it a through the first several weeks, let the lawn grow to seed. Then cut it and feed to chickens or let the chickens on the lawn after it's grown to seed to get all sorts of insects and seeds and then cut and incorporate the lawn as feed. It would double as bedding in the coop and run and eventually triple it's worth as compost. Grass is essentially hay, just not grown out fully. After you cut it, it will grow back. In my growing season here in the midwest I would likely get 3-4 cuts in a year. I live on 1/3 of an acre and over 1/2 of it is grass after you deduct the house, patio, driveway, shed/coop, run, flower beds and garden. It's a lot of grass for our small flock. As a bonus, the grass is just beginning to grow again as we are getting of winter cold and the rains are coming in.
I'm also planting the garden soon. We just started a bunch of seeds indoors yesterday actually. So in just about two months we will start to get some harvests. A particularly useful crop in the situation described in the first post would be radishes. Radish greens could be feed to our birds and the roots would help feed us. Radishes grow very fast. Another rapid growth option would be sprouting grains. If you cannot buy grains, you have a lawn full of grass and if you grow it to seed like I said above you could take some of those seeds and grow sprouts. Sprouts are nutritious and could be fed to the birds and us. All the while, growing other vegetables and fruit would provide variety and added nutrients in time for both birds and us. If successful in meeting the chickens feed requirements, they may be able to reproduce (assuming there's a rooster in the flock, which I don't have but I do have a drake in my duck raft), providing a source of meat for us. It's amazing what one can accomplish on a small plot of land.