when I was 12 ---1957--I raised 100 chickens ---50 hens 50 rooster---with 4H----used grain ---but they were kinda free range on a pen ---can't help with the compost idea but am willing to help if I could------one thing I did find is that if they do not get enough calcium they will eat their own eggs-----adding calcium stopped that------another problem was when my egg production went down I also discovered by head count that some chickens were missing----a raccoon was having a feast ---live trap caught him and we moved him elsewhere ------good luck ---

the New Haven County agricultural station helped me with their guidance

I've read where some grind up the egg shells very fine and feed that back to the chickens but I would think that the Law of Diminishing Returns would kick in as the amount of calcium available would decrease overtime.

obv this is a city thing. but if i wanted to do it, there are restaurants and a fruit market w/in a couple blocks.

since chix are omnivores, meats scraps can bridge that gap.
both dog and cat food are fine, too. cheap dog food is about the same protein as regular feed. better grades are about broiler feed level.
if i have boys i plan to eat, i pen them together and raise them on dog food.
if i feel the girls arent getting enough protein, like they're picking feathers, i give them a little cat which is the highest pro, about double usual feed.

some folks raise maggots for their chix. black soldier flies are the preferred species, but you could rig some bait like a bit of old meat, and let common flies lay eggs.
i sometimes bring up worms for them. a piece of plywood or some such, either in a wet spot or after a rain, or run a sprinkler then turn it over. i have some big logs that i roll over now and again.
i'm sure there are other strategies for insect protein.

They can't free range and a compost pile not too hot for bugs to survive may freeze.

Fish can be an option and there's no size or possession limit for suckers and can be fished year round.

Another thing I'll need to research.

there's lots of dyi set ups out there, and it's simple to collect the larvae.

As they require warm conditions.

Winter squash and pumpkins would work as some varieties can keep for a long time and chickens love them

you could use a seed mat, or other direct heat.