Preheat the oven to 350 degrees. Remove the stems from the cremini mushrooms and set aside. Remove the stems from the shiitakes and freeze for another use (such as making soup stock).
Place the cremini caps, stem side down, on a baking sheet and bake until they exude most of their liquid, about 15 minutes. Remove, drain off the juice and set aside.
Chop the cremini stems, the shiitake caps and the morels. Melt the butter in a medium skillet over medium-high heat, and when the foam subsides, sauté the chopped mushrooms, shallots, garlic and thyme until golden, about 8 to 10 minutes.
Remove and place in a bowl. Toss in the parsley and all but 2 tablespoons of the cheese. Season with salt and pepper to taste.
Turn the cremini mushroom caps over and mound the filling in the caps, pressing gently (there may be a little stuffing left over; it's great tossed in pasta or on a pizza). Sprinkle the remaining cheese over the stuffed caps and top with a sliced pickled pepper. Bake until the mushrooms are tender and the stuffing is golden, about 15 to 20 minutes. Serve warm.
Nutrition information per mushroom cap: