But you might be able to grow plants outside the run, that will stick in through the wire and the chickens can eat just those parts. I'm not sure what does well in shade, though.
You could try peas-- any of the kinds that people grow in a garden for people to eat. They grow nice vines, and the leaves/stems are safe to eat too. So they can climb up the outside of the fence, the chickens can pick bits off through the wire, but the chickens cannot scratch the roots up. Or beans, the climbing/vine kind. Those leaves/stems are edible too. I'm not sure how well either one does in shade, but it only costs 1 packet of seed to find out (unless you have seed leftover from planting a garden--then it's free.)