Marc,
If you like growing shiitake mushrooms then I think you will also like growing Wine Caps. They are very easy to grow. They do have a couple of differences from shiitake, namely that they really need some ground contact and they actually like having a bit of sunlight. Actually, Wine Caps are pretty bulletproof and therefore a great starter mushroom--but you are past the starter stage.
I have a couple of threads I keep running and updated Here:
https://permies.com/t/82798/composting-wood-chips-chicken-litter
And HERE:
https://permies.com/t/130092/mushroom-newbie
The first documents my journey from being a complete novice to having a degree of competence. I usually include this link for people who have never grown mushrooms before so this may or may not be helpful to you.
The second includes step-by-step instruction for a person who was growing under some pine trees. If you leave out the pine tree part, it is a pretty good resource for growing Wine Caps.
Everything I have mentioned is specific to Wine Caps, but Blue Oysters are just as good at breaking down wood (maybe better?), so they may be a fine choice as well. But Wine Caps are just not all that picky and therefore are a great all-around mushroom, especially for making compost (which is magically fertile BTW).
Welcome to the Wine Cap bandwagon!
Eric