Break cauliflower into small pieces and place in a large pot with 3 cups of broth.
Cook until cauliflower is tender, then transfer mixture to a blender. Blend until smooth and set aside. If you have trouble blending it, you can add more of the broth a bit at a time until it blends easily.
In the pot, add oil, leek, mushrooms, garlic, and tarragon if using. Sauté on medium heat until leeks are soft, 5 min or so, then add blended mixture, salt, and pepper. If soup is thicker than you would like it, add more broth until desired consistency is reached.
Bring to a simmer and cook for 20 – 30 minutes.
Serve with a drizzle of oil, a sprinkle of tarragon, and some parmesan if desired.