Minimum commercial feed required?

My chickens always act like they're starving. I've read out on the webs that chickens generally eat 3-4oz/day (of mash, and I think that assumes that's their main intake).

They don't free range; they're in a run that's about 600 sq feet at a time, and which moves every couple-few weeks. Right now the grass is dry and there's not much self-forage.

I'm feeding my 13 chickens (12 right around 5mos old plus 1 adult hen) an organic mixture of grains/legumes (shifting from grower, 15% protein - to layer 20%; right now I'm mixing them around 50/50). I ferment it, b/c in theory that's supposed to improve bioavailability and help them eat less (which I don't see, though I generally only ferment for 1 day, sometimes 2, if that's relevant). I don't leave food "out" all the time, so nobody else (e.g. rodents) is eating it, and they always finish it. I feed them several pounds/day of produce from my garden (the grass in their run is brown since no rain, so that doesn't contribute much). They can dig around for bugs, although with the exception of 1 or 2 of them, they are generally not super fond of worms (annoying!).

SO, my question. It SEEMS like given how much produce I give them (currently apples, shredded zucchini, kale/broccoli leaves), and that I ferment their feed, I should be able to give them noticeably less than 4oz each per day. When I tried 2.25 pounds total (2.75 oz/bird), because of the large quantity of produce (and also I had a half gallon of expired milk that I fed them over those three days to supplement protein), they "acted" like they were starving. They certainly weren't, they just ate frantically when I dished it out. So I bumped that back up to 3.25 oz/bird, and they still act quite hungry (eating super way fast when I show up with food). So I've shifted that back up to 3.75 oz/bird, which they still finish readily, but they act less panicked about it. It just *seems* like so much, based on how much other food they're getting. I realize the produce is less calorie-dense. Is it really just barely impactful?

I'm working on building a compost pile for the run they're about to move to. The last one, in their prior location, wasn't broken down enough to be providing lots of bugs before I moved them, so I do not have this angle dialed in yet...

Besides wishing I could keep feeding (and thus per-egg) costs reasonable, what I'm really wanting is to reduce reliance on outside feed, I want to increase reliance on feed I can generate on my land (turning their beaks up at worms is not helpful). For the winter: I have maybe 80-90 starts of kales and mustards that I'll be growing as a chicken garden for winter-food (I'm zone 8, maritime PNW, so this should hopefully grow enough before it gets cold, and overwinter), plus I grew some extra winter squash to share with them.

How much feed would you think is necessary in this scenario?

I'll start by saying I always free feed my hens they eat what they need. I only don't do that if I get a rodent issue.

Zucchini gives basically 0 calories just like it does for us. kale and apples are also low in calories. I found someone mentioning that;

White Leghorn's with a body weight of 1.5 kg (3lb) needs
   177 calories for body maintenance
   276 calories at 80% egg production.

Your feed might be different but I found various figures online giving an caloric amount of about 80cal per ounce of layer feed. so if your chickens are light weight egg layers then to meet their requirements you would need to feed 3.5oz right in the middle of your comment at the top, if they are a heavier breed it would be more. ok so if you wanted to drop that to 2.75oz you have a shortfall of 75cal per bird per day which would add up to 12lb of zucchini which is an awful lot of zucchini! or 4lb of kale or 4lb of apples every day. this would really throw out the protein content though. If they are not laying yet then wow those are some greedy hens and they don't like worms? weird things, mine loved them and frogs, but not slugs and mice they would all just sit up on the perch and watch, they reminded me of a row of old women standing on a bench holding their skirts up!

I did raise 2 clutches of muscovys and 5 egg layers on home made feed. It was a mix of black oats, rye, barley, alfalfa pellets(waste of time they left the mush), potatoes and pumpkin.  But it was over summer and they could free range. I spent a lot of time working out the protien etc %s and then the birds spent a lot of time picking out the bits they liked and messing up all my work.

Thank you, Skandi. That gives me some really good thinking material. They are varying breeds and non-breeds. Mainly they seem not that big except a couple. There is 1 adult hen who lays almost daily, and 2-3 pullets who have begun laying. I think what I might do is free-feed them dry food with little extra (except gardening weeds) and measure how many pounds they eat in a 24-hour or 48-hour period, and then repeat that with fermented. I can probably find out the # of calories per pound from the grower, and that will give me a target of flock-calories as a baseline. At least I'll have a starting place for what happens when I give them produce, how much to jigger their feed by (then it'll get cold or they'll molt and throw everything off, lol). They probably do get several pounds of apples when apples happen (b/c lots fall off the tree!), but you help me understand that the kale and zucchini, while probably several pounds combined per day, possibly really is just too much water to be noticeably offsetting feed.

Hilarious observation about the mice! (I personally would totally be with them). I have a couple flock members who enthusiastically eat worms. Others I literally see spit them out!! I have hopes that the non-worm-eaters might eventually learn by observation...