I wasn’t involved in the bed preparation, but from what I observed the beds were simply damp sterilized straw.
They produced hundreds of tons per month that way, in cinder block rooms of 7 tiers of wooden frame beds, 3 stacks per room, about 150ft long. They were grown in complete darkness, button mushrooms apparently benefit from that, but oyster mushrooms required light stimulation of fluorescent lights.
Preparing mycelia culture requires biological laboratory techniques, flame sterilized tools, autoclaved culture media, and a laminar flow HEPA filtered booth in order to work reliably, otherwise it will just be mold and random fungus cultured. Or commercial spawn can be obtained easily, and could be multiplied by innoculating sterile media.
I have grown wild oyster mushrooms with some success by piling their preferred food here, balsam poplar, in shady spots in the forest. The trouble with outdoor mushrooms, especially oysters is that they are always infested with fly grubs that seem not slowed down by refrigeration. Rapid freezing or processing solves the problem, and the bugs taste just like mushrooms.
Oysters grown outdoors seem very picky, hard to predict. They are very specific to temperatures and seem to only emerge after soaking rains. They will emerge from the same logs for several years, so don’t give up hope on your project yet.
Shiitake can be grown similarly on oak logs innoculated with wooden plugs. The innoculant can be multiplied on fresh sawn oak sawdust.