No plants really are chicken proof, except some herbs they don't like and don't eat, but even those they will dig up and scratch up. Most chickens will damage most plants (other than already-well-established trees, shrubs and perennials) if they get the chance. Some plants the chickens will eat. Others they'll kill, bit by bit, by continually digging, scratching and dusting around the roots. So, think this out carefully. Personally, I'd plant shade plants outside the chicken run and put shade cloth over the top of the chicken run. If you must plant inside the run, put down weedblock fabric and plant your chosen plants (hopefully wooden shrubs they won't be as inclined to eat) through holes cut in the weedblock fabric. Then, use stone or rock mulch, and the stones/rocks need to be big enough the chickens won't kick them out of their way. Between the weedblock fabric and the stone/rock mulch, you should be able to protect the plants' root zones from the chickens. Of course, you don't want the whole chicken run covered in stones.

We worked around this issue by building our chicken coops beneath tall trees that would provide shade for much of the day. Anything we've ever planted near the runs has been planted outside the runs and far enough from the runs that the chickens cannot reach through the fence and nibble the plants.

Much would depend on how much time the chickens spend in the run. The less time they spend in there, the less likely they are to mess with any plants inside the run. I sometimes let my chickens free-range go inside the fenced veggie garden (which actually is a combination of veggies, fruit, flowers and herbs) but only at selected times of the year (like when grasshoppers invade) and I expect to lose a certain percentage of plants to them. They generally won't bother large flowering annuals and perennials that are tall and well-established, but will destroy those same plants when they are small. So, timing is everything. Chickens are not allowed in the garden as long as it has any leafy crops like kale, Swiss chard, lettuce, mustard, collards, etc. because they'll attack them vigorously and devour even very large plants in no time at all. The concept of growing something inside their coop for them to eat is a nice one, but totally unworkable because they'll devour the tiny seedlings before they get a chance to grow. If you have two separate runs, or a run fenced into at least two separate areas, you can plant one area and let the plants grow to a pretty decent size before allowing the chickens access to it. Then, while they are enjoying the plants you grew for them, sow seed in the other run/other section of the run and let those grow for 4-6 weeks before you allow the chickens back into that area.

Because my chickens devour every bit of green in their run the minute it sprouts, I don't grow things in the run for them. I do, however, grow a huge raised bed of greens for them inside my fenced garden. The fence is 8' tall, and as far as I can remember, they've never flown over the fence. They have sometimes come in through the gate if I am stupid and forget to close the gate to exclude them. I don't let the chickens come into the garden to eat the plants, would be the equivalent of letting a fox into the chicken coop with the same predicatable results. I cut the greens using the cut and come again method and toss them into the chicken run. Using this method, our chickens have kale, lettuce, etc. from March through June or July at the very least (it depends on how early we get really hot). Sometimes the chickens get too excited and find ways to dig under the garden fence (particularly the banties) so they can eat all the greens they want. When that happens, I chase them out, fix the place where they came under, and watch them carefully to make sure they don't repeat it. For the summer time, I grow Armenian cucumbers on the garden fence for the chickens. I let the Armenian cukes get hugely oversized so that they become more melon-like than cucumber-like (not difficult since they actually are melons that resemble cucumbers while small). When I harvest Armenian cukes for them, I just slice the cukes in half like a big long melon and put them on the ground and the chickens devour them. The hotter and drier it is, the more quickly the chickens eat the Armenian cukes. Usually, we have enough Armenian cukes to feed them to the chickens daily in July and August, and sometimes well into September.

So, I think you can grow things for your chickens and grow things to feed your chickens, but in order for them to grow large enough to provide an actual meal for the chickens, they first must be protected from the chickens for a substantial period of time. When chickens see green plants growing, they don't ask "May I?". They simply take what they want when they want it, and they aren't shy about doing so. Mine will eat green onion leaves (in fact, this is one of their favorite things), so even the onions aren't safe from them.

Hope this helps.

Dawn