I've done it before. Considering how long it takes them to sprout, starting them in more controlled environment is attractive. On the other hand, the seeds can be sowed very early outside --- it just takes as much as 3 weeks for them to sprout and you have to make sure to keep the seedbed from drying out. Some people recommend putting a piece of wood/board on top of the soil over the seeds initially.
You can't wait too long to transplant -- no more than two true leaves ...and by then the root can be 2 inches long. The single delicate taproot is what turns into the carrot, and breaking it or even the root finding the bottom of the tray could/will result in kinking or branching.
To transplant, I found it easiest to either remove with cushion of soil cradling it *or* completely bareroot it by floating and rinsing in a bowl of water. Make a deep hole so as not to let the taproot get bent and wide enough to see/maneuver, hold the seedling up against one side of the hole and fill in, making sure not to bury the crown.
...very tedious. Not sure I will try this again unless I can come up with/find a better technique