Wow, congrats on your first egg!
I would just offer oyster shell on the side until ALL girls are laying. And you don't even have to switch to layer if you don't want. In fact Faverolles are dual purpose chickens and 16% protein may be too low long term as it's intended for light bodied layers like leghorn.
Because most of my birds are dual purpose in addition to having chicks, pullets, roosters, layers... I use flock raiser with 20% protein and offer oyster shell on the side. I never use layer. It (flock raiser) is acceptable for all my birds and simple to feed everyone the same.
Please note that laying is NOT effected by the cold... but the hormone that tells them to lay is effected by the amount of daylight hours. It just happens to get colder as we lose daylight.

Girls coming into lay this time of year are often a little delayed. In fact, faverolle tend to be slower developing than other breeds, so that would be surprising to me if it was her that laid. Though ALL are individual and anything is possible!

Everybody should have pullet sized eggs, though there may be occasion when they first start laying that they get double or triple yolkers from the reproductive system working out it's kinks. And those will seem huge! What other girls do you have?