Preheat oven to 400°. Pulse sunchokes in a food processor until coarsely chopped, working in batches if necessary. Spread sunchokes evenly on 2 large rimmed baking sheets. Drizzle each sheet with 1 tbsp. oil and sprinkle with 1/8 tsp. salt; toss well. Roast sunchokes until tender, about 20 minutes.
Meanwhile, heat 1 tbsp. oil in a large pot over medium heat and add shallot; cook, stirring frequently, until softened but not browned, about 2 minutes. Add garlic and cook 1 minute more. Stir in coconut milk and vegetable broth, increase heat to medium-high, and bring to a simmer.
Heat remaining 2 tbsp. oil in a large frying pan over medium-high heat. Cook brussels sprouts, stirring occasionally, until softened and starting to brown, about 4 minutes.
Transfer roasted sunchokes to pot of broth and blend, using an immersion blender (or blend broth with sunchokes in a blender) until smooth. Adjust consistency with water if needed and season to taste with salt.
Ladle into 6 bowls and top each with some brussels sprouts, hazelnuts, and a thin drizzle of oil.