I've done the same as you, plugged up some oak logs and left them outside in a fairly sheltered spot. I also got an indoors kit (spawn and substrate) which is fruiting well and requires very little effort.How do you grow the mushrooms? Did you just get a kit with all the growing medium etc? It's something i've been wanting to have a go at for a while, just not got round to it yet.
I want to grow lions mane mainly. I did buy some plugs a few years ago that were drilled into an oak log. It was covered in black plastic bags and left in the cellar for a while until the mycelium colonised the wood. Which it did quite quickly, the entire log was almost white with it.
Then i 'planted' the log outside in the shady area of garden that never really sees direct sunlight, and nothing happened, the mycelium disappeared after a short period and i didn't get anything at all from it. Quite frustrating really.
I watched a couple of videos on youtube about growing ginger from the supermarket and it seemed straight forward enough, but needs a greenhouse i think, or a big windowsill of which i have neither.
Shame the lion's mane didn't fruit. I'd like to try growing it too. I seem to recall reading it does well on smaller limbs. I think it's fairly slow to colonise.