Just for turning over you plant most many herbaceous plants and veggies, but I think beans a mix of beans and some other greens would be good.
Well, I should caution that you would want a plant which will not self-sow easily or that at least you can take away most of the seeds/seed pods, since you don't want them growing up in the following year when you are trying to plant other veggies.
Otherwise, composting it by burying some mulched leaves and grass would be OK. It wouldn't have to be done just at one time either, you can add to it as the growing season continues.
An alternative to the regular compost method (or mulched leaves) would be where you would bury kitchen wastes within the area. Dig down to about 1 foot, fill it up with wastes, cover with the dirt that you dug up. Proceed down the line to cover the entire 20x20 area, and restart again either slightly offset from the original row or maybe in the perpendicular direction.