2. Rub the new potatoes with olive oil, season with salt and pepper and roast them in a 350°F oven until cooked through, about 35 minutes. Shake the pan from time to time for even cooking. Remove the potatoes from the oven and set aside to cool.
3. Cook the broccoli florets or green beans in boiling, salted water until tender. Drain well and refresh with cold water. Drain again.
4. Bring a large pot of salted water to a boil for the pasta. Meanwhile, cut the potatoes into bite-sized pieces. Remove pesto from refrigerator. Heat 2 tablespoons olive oil in a large saute pan and cook the potatoes until golden on all sides. Drop the pasta in the boiling water and stir well. Add the broccoli or green beans to the potatoes and warm through, stirring well.
5. Turn off the heat and when the pasta is done, drain it, reserving a few tablespoons of warm pasta water. Add half the pesto and the pasta water to the vegetables in the pan, then toss well with the drained pasta. Serve at once and pass the cheese at the table to sprinkle on top. (Store the remaining pesto in the refrigerator for up to 3 days.)