Heat the olive oil in a large pot over medium heat. Cook and stir the leek in the hot oil until translucent, 5 to 7 minutes. Stir the sunchokes, celery, carrot, and garlic into the leeks. Season the mixture with salt and pepper. Add the bay leaves and 1/2 bunch chopped tarragon. Increase the heat to high; cook and stir until the vegetables are hot, 2 to 3 minutes.
Pour the vermouth into the pan; bring to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon. Add the vegetable broth, reduce heat to medium-low, and bring the mixture to a simmer; cook until the vegetables are completely tender, about 30 minutes. Reserving a few sprigs for garnish, add the watercress to the soup; allow the watercress to wilt in the hot soup, about 5 minutes.
While the soup simmers, melt the butter in a skillet over medium heat. Add the mushrooms to the melted butter, season with salt and pepper, and cook until completely tender, about 5 minutes. Stir in the remaining tarragon and set aside.
Remove and discard the bay leaves. Reserve about 1/3 of the soup in a separate pot. Pour the other 2/3 of the soup into a blender in batches, filling the pitcher no more than halfway full. Holding the lid of the blender with towel, carefully start the blender, using a few quick pulses to get the soup moving before leaving it on to puree. Puree in batches until smooth and pour into the pot with the unblended portion. Serve in bowls, topping each with a large spoonful of the cooked mushrooms and a sprig of watercress.