Preheat the oven to 475°. Melt the butter in the oil in an enameled cast-iron casserole. Add the shallots and thyme, season with salt and pepper and cook over high heat, stirring, for 2 minutes. Stir in the white and shiitake mushrooms. Bake for about 15 minutes, or until the mushrooms are just wilted. Stir and bake for about 10 more minutes, or until the mushrooms are dry.