Clean the Chicken of the Woods mushrooms thoroughly with a brush or damp cloth. Chop into bite-sized pieces.
Heat olive oil or butter in a large pot over medium heat.
Add diced onion and sauté until translucent (~3-4 minutes).
Stir in minced garlic and cook for 30 seconds until fragrant.
Add carrots, celery, and chopped mushrooms. Sauté for 5-7 minutes until mushrooms soften.
Pour in the broth, thyme, and smoked paprika (if using).
Bring to a boil, then reduce heat and simmer for 20 minutes.
For a creamier texture, use an immersion blender to partially puree the soup.
Stir in heavy cream (or coconut milk) and simmer for 2-3 more minutes.
Season with salt and pepper to taste.
Ladle into bowls and garnish with fresh parsley.