No need to till.
You don't need to wait 6 months, or even 6 minutes. You can do as Puppies said, and if you want to plant right away, do the following:
-For seeds: Using a trowel or your hands; dig a 1-2 inch wide trench all the way to the cardboard, and then make a cut in the cardboard to allow your seedlings
roots to penetrate
-Or you could try pre-slicing the cardboard before you bury it in compost and mulch, and planting your seeds in the compost, above where the cuts were made. \
-For transplants: do a sheet mulch, pre- cut X's in the cardboard about 4 inches long at even spacing. Plant your transplants above the area where the X's are. It doesn't have to be bang on, just within a few inches. The roots will find their way through the cardboard.
**Unless you are piling more than 6 inches of heigh of compost on top and/or your existing soil is so loose you can plunge into it down to your wrist; I'd stay away from
root crops in the first year, unless you want to plant them as companions, to do a deep tilling for you. You still may end up with a harvestable root crop but don't count on it. I've seen this succeed, and I've seen it fail. Radish or baby carrot varieties, or even parsnips might work.