How big is your chicken run that you compost in?
My chicken run is about 13X39, open to the sky, with bird netting on top for protection.
I scoop poop each morning into a bucket. When the bucket gets full (usually 2-3 days), I dump the bucket on one of my over-winter raised garden beds, or in the large (20x50) chicken run (if the veggie garden is in use).
I use a dry deep bedding (mostly shredded paper these days) and clean out everything about twice a year. All my used coop litter gets dumped into the chicken run composting system. I rotate areas where I take out the compost. At this point, all the compost I remove is at least one year old.
I also throw leftovers, veggie scraps, leaves, hay, weeds/grass clippings into the run.
Same here. Although my chickens normally eat almost everything from the chicken bucket waste scraps and leftovers in only a few minutes every morning. The bulk of my chicken run compost is leaf mold, grass clippings, and any other material (weeds) I might toss in during the year.
I top off each garden bed with a layer of dirt compost (usually my neighbor buys a big load and lets me take a little for the top off in exchange for my help for the day).
According to the owner of the local nursery where we get some of our plants, he recommends mixing a high-quality topsoil with my chicken run compost 1:1. Compost is great, but it does not have the same stuff that good topsoil does. And topsoil does not have the same stuff as compost. Together, they make magic, I guess.
I live on a lake and have very poor native sandy soil. I have to buy my topsoil by the scoop of a Bobcat loaded into my 4X8 utility trailer. For years I tried to amend my native sandy soil with compost. Sure, it improved the soil to a degree, but when decided to use raised beds with quality topsoil and compost, it was like night and day difference for me. That was the magic trick for me to grow food.
I have found though that although my run is constantly being turned by the chickens, it is too “chunky” to take any compost from the run to add to the beds. I am pretty much on my own w this project, no extra machinery, but I was wondering if you could think of any suggestions to help it break down faster.
I have moved to using shredded paper as my dry deep bedding for the last ~3 years. Paper shreds breakdown much faster than other litter I used in the past. When dumped into the chicken run, the paper shreds coop litter is compost ready in a few months. I usually let it sit longer, of course, because the chicken poo needs to "cool" down before being put on a raised bed garden.
With your 20X50 chicken run, and if you are dumping all your organics in there, I suspect you could be harvesting as much compost as you need and barely scratching the volume in the run. Well, in my case, I harvest hundreds of dollars of compost every spring for my raised beds and I estimate only using less than 10% of my compost in the run.
As I said, the bulk of my chicken run compost is leaves and grass clippings. At the front of my run, the compost litter is about 18 inches deep. The chickens are constantly turning that litter and I usually harvest the compost at the back end of the run where it is about 6 inches deep. By the time the compost litter gets to the back of the chicken run, it's really broken down well.
But I always sift my compost litter just the same. For years, I used a simple 2X4 wire frame on top of my wheelbarrow for sifting out my compost, like this...
That worked for many years. The finished compost sifts out into the wheelbarrow, and I tossed the larger stuff back into the run for more composting. I always had enough sifted compost for my needs.
Of course, if your goal is to break down the compost faster, then you might need to manage the compost litter a bit. Like ensuring that there is enough water in the litter to keep it at that wrung out sponge consistency. Maybe you need to turn that litter some on your own with a pitchfork or tiller. The chickens do a great job on their own, but it never hurts to go deep with a pitchfork. Mixing in a lot of green grass clippings will heat up the compost litter and speed up the breakdown process as well.
In my case, the chicken tilling and nature is fast enough for the amount I remove every year. My success is really built on having so much material available that I never worry about how fast the compost litter is breaking down. I would estimate, at this point, that all the compost I harvest from my chicken run is over 2 years old.

Sifting all the compost I used every year was a lot of work. I decided to upgrade my system and invested in a cement mixer compost sifter setup...
At the time I built that cement mixer compost sifter setup, I calculated that I could sift out about $60 worth of finished compost per hour with little manual effort compared to the old sifting frame on the wheelbarrow. In less than 4 hours, that setup paid for itself. Since then, I have more than doubled my raised bed gardens and have sifted out hundreds and hundreds of dollars of finished chicken run compost every year.
The finished compost sifted into that black wagon gets mixed with topsoil and put into my raised garden beds. The rejected material in the gray wagon, too big for finished compost, gets tossed back into the chicken run for more processing or I use it as top mulch in the garden beds as a sun/weed block.
The sifting process helps break down all that material as well. Even if not small enough to sift out as finished compost, the sifting process breaks down the rejected litter and will speed up the process for the next turn in the chicken run.