I do not advise trying carrots until your third year of digging the soil loose, but I've had good success with them for many years. THIS winter Fred and I had carrots for every home dinner from January into April! I dig them weekly, which this year was more of a project than before.
Each spring I scatter seeds on a sizable part of my garden and then cover them with floating cover so birds and bugs don't eat them. This year, for the first time, wild winds tore up the FC. I replaced it, but soon noticed that weeds were growing abundantly under it. It's not easy to replace, so I ignored the problem until after Saturday's open garden. On Sunday I saw not only lots of weeds, but 2" carrot seedlings in tight bunches. Apparently the carrot seeds blew around too, along with the weed seeds. I started taking out the weeds and started transplanting the seedlings into the many large empty spots.
It seems to be working! Transplanted carrots have never grown well before, but this weather is ideal. They get plenty of moisture! I'm keeping my fingers crossed that I'll have the usual abundant carrot harvest this winter.