Your coldframe project is giving me ideas about making something like this to put on top of my raised bed where the carrots are. I plan on harvesting the carrots before it gets too cold this year, but the cold frames do help to warm up the ground in the spring so that I can get an earlier start on planting whatever I end up planting in the raised bed next year.
I would suspect if you put a cold frame over your carrots, you wouldn't have to harvest them all at once. When the days get shorter they'll stop growing but will be ready to pick as you need them!