Preheat the oven to 400°F. Wrap the garlic in tin foil, drizzle with olive oil, and roast until very tender and fragrant, about 20 minutes.
Remove the roasted garlic cloves from the head and toss with the walnuts, dill, sugar, walnut oil and vinegar in a small bowl. Adjust the seasoning with salt and pepper.
Meanwhile, melt the butter in a large saucepan. Add the onion and garlic and cook on low heat until softened but not colored, about 5 minutes.
Stir in the thyme, mushrooms, broth, brandy and cream. Bring to a simmer and cook for 20 minutes, until the mushrooms are tender.
Purée the soup with an immersion blender or a regular blender (let it cool first a bit to avoid splashing). Season with salt and pepper.
To serve, divide the soup between bowls, garnish with walnut-dill relish and serve with baguette on the side.