Melt the butter and heat the oil in a pan over medium heat. Add the onion and saute until tender, about 5-7 minutes. Add the garlic and thyme and saute until fragrant, about a minutes. Add the cremini mushrooms and saute until the start to caramelized and turn golden brown, about 10-15 minutes. Add the wine, deglaze the pan and cook until most of the liquid has evaporated. Season with salt and pepper and remove from heat and let cool a bit.
Mix the cheese into the mushrooms. Butter one side of each slice of bread and place 2 in the pan buttered side down. Top each with 1/2 of the mushroom mixture and finally the remaining slices of bread with the buttered side up. Grill until the cheese has melted and the bread is golden brown, about 2-4 minutes per side.
