Preheat the oven to 400°F. Scatter the walnuts on a baking sheet and toast in the oven for about 3 minutes, or until fragrant. Take care not to burn them. Set aside, dividing half for the pesto and half for serving. Toss the carrots in 2 tablespoons olive oil and scatter on the same baking sheet. Roast the carrots for about 10 minutes, or until the color has deepened and they are slightly browned and tender.
Place the roasted carrots, 1/2 cup walnuts, garlic, basil, Parmesan cheese, salt and pepper in the bowl of a food processor, and process until smooth. With the food processor running, add the remaining olive oil slowly, until it is well blended. If you desire more creaminess, you may need to use a little more olive oil. Season to taste with salt and pepper if necessary.
Meanwhile, in a large pot, cook the farfalle pasta in salted water according to the package directions. Drain, reserving about a cup of the pasta water. Transfer the cooked pasta to a large bowl.
Dress the pasta with the carrot pesto, and toss well until it is evenly coated. Use some of the reserved pasta water to make it easier to coat the pasta, adding in small increments. Season to taste with salt and pepper and serve with additional toasted walnuts and shaved Parmesan cheese. Serve immediately.