For a quick start .. bung some KY gell( A desset spoon or more ) in a corner of a soft poly bag and add all your seeds you intend sowing , mix them in gently .
Bung them in the airing cupboard and they should have started to sprout by day four .
Add a bit of water to the bag , make a balloon , shake gently to free up the seeds and dilute the gel a bit , snip off a small corner to make a 2 mm hole then gently squirt the pre chitted seeds direct into a dampened seed line .
do it as thinly as you can .
Cover with sifted dry sand/soil to a depth of 3 mm .
This should cut the growing time for the carrots by at least a fortnight . I say KY jel for it's easy to get and is sterile but does not contain any fungicides or preservatives that would kill of the seed germinations.
If it is dry keep watering the sownings. don't thing the young plants in day time do it towards the evenijg if you must thin them as carrot fly will seek out the plants , don't leave thinnings in the area or on the open compost heap for the same reason.