Light the grill or make a charcoal fire. Bring 4 quarts of salted water to a boil in a large pot for cooking the pasta.
Remove and discard the green stems and the fronds from the fennel. Remove any blemished or tough layers from the fennel bulb but do not trim a thin slice from the base (the base will help keep wedges intact). Slice the bulb crosswise from top to bottom into 1/2-inch-thick wedges.
Peel and quarter the onions, making both cuts through the top and bottom ends on the onions. Brush the fennel wedges and red onion quarters with 2 tablespoons oil. Generously season the vegetables with salt and pepper.
Use a stiff wire brush to scrape the hot grill clean. Grill the fennel and onions, carefully turning them once, until the vegetables are marked by dark stripes and are tender, about 15 minutes.
Briefly cool the grilled vegetables. Cut the fennel and onions into thin strips. In a large bowl, toss the grilled vegetables with the sun-dried tomatoes as well as the remaining 2 tablespoons oil, the vinegar and the basil. Taste for salt and pepper and adjust seasonings if necessary. The vegetables can be covered and set aside for up to 1 hour, if desired.
Cook and drain the pasta. Toss the hot pasta with the grilled vegetables. Mix well and transfer portions to warm pasta bowls. Serve immediately with grated cheese passed separately.
Best pasta choice: Fusilli or other short, curly shape