You can supplement with raw eggs, cooked eggs (scrambled or boiled, no seasoning). You can just smash up the entire egg, shell and all, and feed it back to them, raw or cooked. This is the best source of complete protein that they need, but they'll need more than you can provide if you're just feeding their own eggs back to them.
For additional protein, any type of low fat unseasoned meat that you might eat yourself is good. I feed mine freezer burned ground beef - I boil it then put it out in the run. I also give them the carcasses of roasted chickens (no/minimal seasoning, especially minimize salt) and let them pick the meat off the bones. Fish or shellfish meat, skin, scales, bones, etc. Whatever is locally available. Insect protein is also excellent, but may not be available this time of year. This can provide temporary protein increase, but won't have all the trace nutrients important for complete health like formulated chicken feed does.
Long term, you'll want 20% all flock feed, whatever is available locally. I like Purina Flockraiser or Dumor chick starter (20% protein). Some chick starter is 20% protein - that is perfect and everyone can eat it year round. Set out supplemental calcium (oyster shell) and grit, and they'll be good.
To break the habit of picking each other, look into pinless peepers and chicken saddles in addition to the advice given by others above. Here's my favorite saddles below - they help feathers beneath it grow back without interference. These saddles last for years, are super durable, washable, and adjustable. These will shield areas on the back, and some of the base of the tail. Worth every penny.