Hi Hmac and welcome to the Forum.
Your hen is still quite a young layer, and they do sometimes take time to settle down into regular laying. Hybrids are often bred for commercial productivity, the egg farmers like a bird who can squeeze out as many eggs as possible in the first couple of years, so what yours is doing is most probably the result of her genes, not anything you are doing wrong. Pullets new to laying do often lay shelless eggs, and so long as the membrane is intact its OK to eat them in any cooking that doesn't require boiling them in their shells. Or you can give them raw to your dog, if you have one - mine loves a raw egg and they're very nourishing. At least she's doesn't seem to have the opposite problem, of being eggbound, which is much more worrying. (The hen, not my dog!) Just make sure she's got access to good-quality layers pellets as well as oystershell and flint grit, provide greenstuff such as cabbage if she's not out at grass, and she will be OK.
Have you wormed them yet with Flubenvet? They should have been done when they arrived, but in any case will need dosing every 4-6 months, to keep the inevitable worm burden under control, as this will help their digestion and gut health. Don't rely in herbal tonics, they won't kill worms. See Poultrykeeper main website, here ; http://poultrykeeper.com/general-chickens/worming-chickens A pack of Flubenvet will worm 20 chickens so will last you quite a while if you haven't got that many chickens.
Oh, and try to remove any eggs ASAP after they're laid, especially the soft-shelled ones, as if the hens find out they can peck and eat them, they will get into a habit which is often impossible to break.