In a small bowl, soak dried shiitake mushrooms in 1 cup hot water for 30 minutes. Drain (discarding liquid), and coarsely chop mushrooms.
In a Dutch oven (or other large, heavy-bottomed soup pot) over medium-high heat, add olive oil. When oil is hot, add hen of the woods mushrooms and sauté until mushrooms just begin to soften, about 5 to 6 minutes. Remove and reserve 1 cup of mushrooms.
Add shiitakes, assorted fresh mushrooms, leek, onion, carrot, celery, potato, garlic, bay leaves, sage, and all but 1 tablespoon shallots and 1 1/4 teaspoons thyme. Sauté until vegetables are tender, about 15 minutes. Remove and reserve 2 cups cooked vegetables.
Add wine and cognac. Cook until wine is reduced by half. Stir in heavy cream and chicken stock. Bring to a boil, then reduce heat and simmer, uncovered, for 30 minutes.
Working in batches, carefully purée hot soup in a blender. Return puréed soup to Dutch oven. Add reserved 2 cups cooked vegetables, stir in lemon juice and season with salt and pepper to taste. Cover and keep soup hot.