Preheat the oven to 400°F.
Place the mushroom caps on a lightly oiled baked sheet and roast cap side down until tender, about 10 minutes.
While the mushrooms are roasting, heat a skillet over medium high heat and melt the butter. Cook the onion, shallot, garlic, celery, red pepper and parsley until softened, about 4-5 minutes. Deglaze the pan with white wine and let cook for another minute.
Add the bread, salt, freshly ground black pepper and thyme. Mix to combine thoroughly. Gently fold in crab meat.
Divide the stuffing between the mushroom caps and place back the baking sheet. Bake until the mushrooms are tender and the stuffing is browned, about 25 minutes.