Anton, you may just be in time to get some short-horn carrots in, but why not try what's worked for me this year, but sowing in a 6" deep planting tray, and kept off the ground on a table-top?
If you have the time - and I have actually done this once with great results, you can try to sow individually, (yeah I know it's a faff, but it's easier than you think), and by rights, you should get some small roots, even a couple of inches or more, and they really do taste good, so may be worth a half hour and some new ordinary compost, preferably with some peat-based as you'll need the moisture retention!