Remove upper stalks from fennel, including feathery leaves. Discard any wilted outer layers on fennel bulb; cut off a thin slice from the base. Wash fennel and pat dry. Quarter fennel bulb lengthwise and discard core; cut quarters in thin strips.
Cook pasta in a large pot of lightly salted boiling water for 5 minutes. Add fennel and carrots; cook for 5 minutes more or until pasta is al dente (tender but still firm); drain.
Meanwhile, in a small skillet cook onion and garlic in hot oil for 3 minutes or until onion is tender. Stir in pepper and salt. Transfer pasta mixture to a serving dish; add onion mixture, radicchio or spinach, and Parmesan cheese. Toss well to mix. Makes 4 side-dish servings.