Combine garlic, basil, salt, pepper and walnuts in a food processor and pulse to combine, stopping machine once or twice to scrape sides with a spatula. Gradually add walnut oil and olive oil, pulsing to combine. Add grated cheese and process briefly. Do not over-grind or pesto will have very little texture. Taste and adjust seasoning, keeping in mind that the cheese adds salt so additional salt may not be needed.

Bring a large pot of salted water to a boil. Add pasta and cook until al dente. Reserve some of the starchy pasta water for thinning the pesto. Drain pasta. Thin pesto with a tablespoon or two of pasta cooking water to obtain a sauce that will coat the pasta. Drizzle pesto over pasta and toss to combine. Top with finely chopped walnuts and grated cheese.