In a cast-iron pot over medium-high heat, warm 3 tablespoons of the olive oil. Add the onion, garlic, and fennel seeds. Cook, stirring occasionally, until softened and translucent, about 5 minutes. Add the celery, thyme, celery root, sunchokes, potato, chicken broth, and 1 teaspoon salt. Bring to a boil, reduce the heat, and simmer, covered, until the vegetables are tender, 25 to 30 minutes. Puree the soup in a blender, and season to taste with salt and pepper.
In a sauté pan over medium-high heat, warm the remaining 1 tablespoon olive oil and the anchovies and cook, avoiding oil splatter, until the anchovies are broken down, 30 seconds. Add the breadcrumbs and toss. Remove from the heat, add the walnuts and chopped parsley, and toss to combine.