Wash all the mushrooms in lukewarm water in a deep bowl. Mix the mushrooms well to loosen the dirt. Remove them from the water by hand, leaving the dirt in the bottom of the bowl. Repeat until there is no dirt. Remove the stems from the oyster and shiitake mushrooms, then slice all the mushrooms. Keep the domestic mushrooms separate from the other types.
Heat an 8-quart sauce pot over moderately high heat. Add the olive oil, the domestic mushrooms, and the minced shallots, and cook until the mushrooms are soft. Add the remaining mushrooms, and saute for 5 minutes, stirring occasionally. Add the vermouth, and reduce for 5 minutes.
Add the milk, fresh thyme, garlic, and nutmeg. Bring to a boil, and lower to a simmer. Cook for 30 minutes, then remove the thyme sprig, and season with salt and pepper. Allow to cool slightly, then puree in the blender.
Meanwhile, quickly blanch the strips of black trumpet mushrooms in salted water. Cool and add to enoki mushrooms and chives.
To serve, reheat the soup in a sauce pot or in the microwave. Divide the tossed Enoki and black trumpet mushrooms into 6 equal portions, and place them in the center of 6 bowls. Pour the soup into the bowls at the table, pouring over the mushroom mixture to heat it up.