I found a big piece of what looks like skin.
I think it might be the "shell" of a softshell egg. That would be a thick membrane with maybe a tiny bit of calcium on the outside to make it stiffer. It would have originally been all around the egg, but got ripped open and the chickens ate the egg. The membrane is a pretty consistent thickness all over.
If the thing you found has lumps or thick parts, then it is something else and I don't know what to say.
But if it is the "shell" (membrane) of a softshell egg, and if it only happens once, you don't need to worry.
If it happens regularly, then you will need to make sure the chicken who produced it gets plenty of calcium, and that may or may not fix it. Apparently some hens have trouble absorbing calcium, but that is less common than hens that just don't get enough calcium in their diet.
I have one hen that is laying and the rest are young pullets that are very close to the general age of laying.
Food - I provide all of them chicken starter at all times. My laying hen gets starter and laying pellet and she chooses what she wants to eat. In the morning i put all the food pales out so they eat what they want, mostly starter.
That sounds fine to me, as long as they also have oyster shell or eggshell or another source of calcium.
Chick starter should be fine for them all.
If they are almost ready to lay, then layer feed would also be fine for them all.
Allowing them to choose chick starter or layer food should also be fine.
I occasionally give them scratch and meal worms for treats. I have been recently giving them watermelon, corn, green beans, celery, and yogurt. They are just now interested in that kind of thing.
If they are having problems, I would stop all of those other things for now.
If they are healthy with NO problems, then feel free to continue whatever you've been doing that works.
In hot weather, you can get some of their chicken food wet with cool water, and most chickens seem to really like it. (No need to limit the amount of wet food they eat, just don't let it sit around and spoil.)
Do you mind elaborating because I thought I finally had the food situation figured out. Why can't they have an occasional treat?
Some people have very strong opinions about feeding ONLY the appropriate kind of commercial feed, and some other people are much more relaxed about it, and some people feel strongly that they MUST feed a selection of other things. Ultimately, you have to be the one who makes the decision for your chickens. But no matter what you decide, someone is sure to tell you that you're doing it wrong.