In a food processor, combine the basil, olive oil, 1/4 cup Parmesan cheese, walnuts, garlic, and 1/2 tsp. salt. Puree until smooth.
Bring a large pot of salted water to a boil. Cook the pasta according to package directions. Drain, reserving 1/2 cup of the cooking water.
Add the chicken and 6 Tbs. of the pesto to the warm pasta. Toss to coat. If needed, stir in up to 1/2 cup reserved cooking water to moisten. Heat through. Top with additional cheese.