it's an ongoing experiment for me. In the spring of last year, I build a portable coop/run setup, with the run being 8x8 and covered. As the chickens destroyed the grass, I threw leaves, grass clippings, etc in there to make a deep litter mulch pile, kind of similar to the OPs "compost run" idea. It ends up being a great bug farm.
About every six weeks, I moved it to a new location, then surrounded the run area with a temporary fence and planted a mixture of field peas and buckwheat. Once the plants were well established, I moved the fence away and let the chickens got at it. I found that they'd generally graze the leaves for a while, then wander on to something else. It helps that they have a couple acres to forage in, much of it wooded. The buckwheat petered out in midsummer, but that's about when the peas started coming it. I picked a few pounds... shelled some to grind or sprout, and threw a few, unshelled, into the run. Kind of fun, watching a chicken bash a pea pod on the ground to get the peas out.
Once the peas started dying, I put the fence back and planted assorted greens in the same spot. Meanwhile, I've planted Austrian winter peas, fodder radishes, mangels, and more greens inside the fenced "real garden". I've been gathering greens for them in a "cut and come again" mode for a couple of months.
With the colder weather, I've left the run in place for several months, and it's become a spectacular bug farm.
Meanwhile, I've started raising meal worms, but that's going slowly, as the temperatures are suboptimal right now.
Plans for the spring include planting comfrey and amaranth.
edited to add:
A local brewery supplies me with spent grain, which I add to the primary feed, which is sprouted wheat, oats and peas (or lentils, depending) toppeds with fish meal and kelp.
In addition to the stuff planted where the run was, I plant clover everywhere, and have take to throwing excess veggie seeds anywhere I think they might grow.
And of course, nature has already provided us with tons of chickweed and dandelions.
And garden/kitchen scraps get thrown in the run as well.