What I've been doing, and it's working pretty well, is attack it on a couple of fronts.
First, feed wet/fermented higher protein feed. I take whole bags of Turkey grower/game bird feed and cover it with water. Within a day or so it "sours" and the chickens love it. They go nuts as soon as I open the barrel because they can smell it (and it does smell). For extra essential oils and vitamins I add in bird seed, which they also love. They extra water in the feed makes dealing with the heat easier on them, and the higher protein makes up for a lack of bugs and things to forage. You may want to ferment smaller batches; you'll want to go through a batch within two weeks.
Second, on the subject of green stuff, I do keep a bale of alfalfa in the run. It last a couple of months, keeps them busy, and provides them with the greens they lack in the desert.
Third, definitely shade. I cannot believe what a difference it makes for them. My runs always have a solid fence to the south to help with shade, then I take pallets and lean them up against the fence. I weave empty feed bags through the bars and they have more shade to get under if they need to.
Fourth, provide them with a shallow pool to dip their feet in if possible. They can cool down quickly by wetting their feet. I place rocks in my little kiddie pool for them to stand on and I see them use it all the time.
Fifth, misters. In the dry desert heat, a little mister system can work wonders and bring down the temps on the hottest days.
Sixth, wet ground. In the morning before it gets really hot on those days that I know will be difficult, I make sure to really soak the ground under the pallets and in other areas where they hang out in the shade. They will dig down to get to cooler dirt and this helps tremendously. They can deal with the heat as long as they can get to cooler ground.
So I hope this helps! I just this summer implemented most of these things and I'm happy with how well they've been doing...including laying regularly...even in the heat. Hubby to install misters soon...