While mushrooms are soaking, heat oil in a large pot. Add the coarsely chopped onion and leeks. Sauté until soft (not browned), three to four minutes.

Clean, slice, and coarsely chop all mushrooms (including soaked drained mushrooms). Add to pot and cover. Lower the flame and let the mushrooms soften 10 to 15 minutes. (If they become dry, add 1⁄3 cup water and cover once more.)

When mushrooms are soft, add wine, garlic, thyme, salt and pepper. Mix and cook over medium heat for five minutes.

Add water plus vegetable stock or water plus milk. Bring to a quick boil and lower heat. Simmer for ten minutes.

Purée soup with an immersion blender until smooth.

Bring soup to simmer again. At this point, you can add the heavy cream, about five minutes before serving.