Preheat oven to 375°F.
Par boil potato and garlic together until fully cooked. Reserve warm as possible thickener for cauliflower.
Prepare steamer. In batches, steam all cauliflower until knife-tender. Place immediately into food processor and puree until smooth. Pass through fine mesh box sieve (for consistency). Add potato if needed to thicken. Season to taste. Keep warm.
Toss apples, butter, salt and pepper, and roast in oven for 15 minutes. Toss and roast 10 minutes more. Keep warm.
Heat a nonstick sauté pan over high heat and add butter until foamy. Add mushrooms, season with salt, pepper and 1 tablespoon tarragon and cook for 3 minutes, tossing. Add shallot and cook 1 minute more. Keep warm.
Place hot soup into bowls. Add apples, chanterelles and remaining tarragon as garnishes.