I've done pit burns where we used the digger to push the excavated dirt over the completed burn. That works well. I also use a thin piece of stainless steel to cover the top of my big kontiki when I want to make a batch of dry biochar. I have to seal the edges with clay.
In either case, be patient and don't uncover until you know it's cooled down. This will take longer than you expect. I've uncovered a batch after four days and had it reignite, even though the kontiki wall temperature was the same as the air around it. Dry biochar is a really good insulator and the interior of a big pile can take weeks to cool down sufficiently.
If the biochar is destined for soil application, then having sand or dirt mixed in is no big deal, and clay is actually a worthwhile additive for biochar inoculation. Have a look at the just-released Farmer's Guide from ANZBIG in the chapter on inoculating...page 121 talks about clay:
ANZBIG Farmer's Guide