Welcome to BYC. I'm really uncertain as to what is going on with your eggs.
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Thank you. I appreciate you taking the time to post.Idk if you know what "lash material" means. If not, be prepared that the following info is very high on the yuck factor. Lash material results from an infection in the oviduct. Lash material is chicken pus. Chicken/bird pus is different than mammals; it is solid, though soft and "cheesy". If you've heard the term "lash egg", it's not an egg at all, but infection that travels through the oviduct and so is somewhat egg-shaped. But a "lash egg" looks Nothing like a normal egg, including it has no shell. It's more unusual for the infection to be INSIDE an actual egg shell, and that is a surprise none of us ever wants to see. The infection (salpingtitis) can be either bacterial or viral. Bacterial salpingtitis can sometimes be rectified with antibiotics, but of course viral cannot. Salpingtitis is unfortunately very common in high-production layer breeds, but again, lash material Inside an egg is fairly rare. Here is an article that will tell you all you never wanted to know.
https://the-chicken-chick.com/salpingitis-lash-eggs-in-backyard/#:~:text=Lash eggs result from an,waxy, cheese-like pus