Where do you live? Potentially how cold are you talking about? If you check out Naked Necks (Turken) you can see that they have no feathers on a fair portion of their head and neck. They supposedly have half as many feathers on their body where they do have feathers and they are considered a cold hardy breed. Some chickens go through a pretty heavy molt in winter and do fine. I don't think you have much to worry about as long as she has a dry place with good ventilation she can go to where she is out of the wind.
If the entire feather shaft came out she should replace that feather fairly soon, probably within a month. If a bit of the shaft was left in she will not replace the feather until her next molt. That could be your next fall.
Is it normal for a hen or pullet to go broody in the winter? It dd not use to be but it's not that rare now. Before they were domesticated they would lay a clutch and hatch them in spring, do it again a couple of times before fall, then quit laying until spring and the weather and food supply got better. We have domesticated them and taught them to lay a lot more eggs each year. We typically feed them well all year long. Artificial lights may extend their day, either on purpose to get them to lay or a security light or street light by coincidence. Some people provide heat in the winter. Any of these things can increase their chances of going broody, especially if it is a hen or pullet that has a strong broody instinct. It is not "normal" for a hen or pullet to go broody in the winter but it is not that rare anymore either. There have been several other threads recently about hens going broody in winter.