Heat the oil in a large pot over medium heat. Add the leeks, parsnips, fennel, ¾ teaspoon salt, and ¼ teaspoon pepper.
Cook, stirring occasionally, until the vegetables begin to soften, 6 to 8 minutes. Add the wine and simmer until evaporated, 1 to 2 minutes.
Add the potatoes and broth and bring to a boil. Reduce heat and simmer until the potatoes are tender, 12 to 15 minutes.
Transfer half the mixture to a blender and puree until smooth. Return it to the pot.
Meanwhile, heat broiler. Place the bread, cut-side up, on a baking sheet and sprinkle with the cheese. Broil until the cheese melts. Cut into pieces.
Sprinkle the soup with the dill and serve with the cheese toast.