I have an EE that lays those ginormous eggs too. I can't put them in a carton, so I lay it in the rack of my fridge and use it for my breakfast. That single egg makes an omlete that is bigger than most 2 egg omletes!
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In her case, she is genetically programmed to lay these eggs. There's nothing you can do about it, and it seems as if you've figured out how to live with this challenging hen! The downside is that, no, she can't keep it up. She's prone to ovarian cancer and other serious ailments. One reason for her weight loss could be that the cancer causes the intestinal tract to become tumorous and often blocked. She might not be able to absorb food or nutrients very well. On the plus side is if she's still laying, then most of her systems are functioning! The hybrids produce a lot, but don't live long. I just lost two Golden Comets, who were also outstanding layers. You can read about what I found on their necropsies here: http://hencam.com/henblog/2012/06/the-kindness-of-euthanizing/