How to Make It
In a large pot, melt the butter. Add the garlic, leeks, celery and onion, season with salt and pepper and cook over moderate heat, stirring occasionally, until softened, about 8 minutes. Add the thyme, sunchokes, potato and wine and cook for 3 minutes. Add the stock and bring to a boil. Simmer over moderately low heat until the potatoes and sunchokes are tender, about 20 minutes. Discard the thyme sprigs.
Working in batches, puree the soup in a blender until smooth; return the soup to the pot. Stir in the heavy cream, season with salt and pepper and keep warm.
In a large skillet, melt the butter in the olive oil. Add the mushrooms, cover and cook over moderate heat, stirring occasionally, until tender and starting to brown, about 7 minutes. Uncover and cook, stirring, until browned, about 4 minutes longer. Stir in the parsley and chives and season with salt and pepper.
In a small saucepan, heat the vegetable oil. Add the paprika and cayenne and cook over moderate heat until the oil has a smoky flavor, about 1 minute. Let cool.
Ladle the soup into bowls. Top each portion with mushrooms and a drizzle of paprika oil. Serve right away.