Carrots are biennial, so they only last 2 years before going to seed somewhere in the 2nd year depending on how warm your spring and summer are.
If your carrots didn't grow big enough to eat in the first year, it's doubtful whether you have enough plant food and water in your soil to get them going further come the next spring anyway. Best to eat the small carrots before the colder weather sets in and start again the next season.
If you do let your carrots over-winter and they start to send out the long flowering stem and leaves in spring/summer, you can either let them flower which will make the bees happy, and save the seeds, or pull them up and juice the now slightly woody roots.