My hens often go broody. Most of them lay in the winter after the molt is over, even in January or February. You can read posts on here where people have hens go broody in winter. It's not common but it happens. I have not noticed if they provide extra light or even heat to their chickens in winter. Mine practically never go broody until the later days of Spring, maybe April but more commonly May at the earliest. I have a lot go broody in the long hot days of June and July.
Hormones trigger a hen to go broody. I don't know what triggers those hormones. Changes to the length of daylight is very important in many hen's laying cycles. Not so much length of daylight as changes to that length. I suspect there are different triggers that cause those hormones to cause a hen to go broody. As I said, I don't know what those triggers are. I suspect actual egg laying may not be one of those triggers. At least not with your hen.
