Sounds like a soft shelled egg broke before the hen (pullet)
was able to expel it. This happened to my hen Joy this summer. I found an eggshell hanging from her vent and like you said, egg contents elsewhere. Is she acting lethargic? Tail drooping? How is she acting? If she is acting lethargic I would go ahead and bring her to a vet if possible. She may have egg contents and shell still inside of her, and if left for an extended amount of time those contents can harden (quickly) blocking her intestines and in Joy’s case, her airways. Is she pooping? If so, that’s a good sign, if it contains a yellow or whitish/clearish substance than that means there is egg contents stuck inside, in which case bringing her to a vet can really save her life. To prevent EYP your vet will try and extract any egg contents, which is something you can do yourself (I’d link an article but I’ll edit my post in a moment) or have the vet do (I’d have the vet do it because you can also take an X-ray, this helped me and my vet figure out that my hen’s airways were blocked) and they probably will give you an antibiotic which you will have to give her once-twice a day. If you don’t think that your hen has any egg contents inside of her than I would continue to watch her.
Soft shelled eggs are actually common with new layers. It’s nothing to be worried about, in fact there are lots of strange types of eggs your hen may lay, and none of it is your fault. If you say they are on Oyster then I wouldn’t worry about calcium deficiency.