In the bowl of a food processor mix the basil, garlic, lemon zest, walnuts, and salt. Pulse until mixture is well combined and a coarse paste is formed.
Turn the food processor to low and slowly drizzle in olive oil. Add cheese, pulse until just combined.
Taste and adjust as necessary. You may wish to add additional seasoning, or additional oil if the pesto is too dry.
Bring a pot of salted water to a boil.
Quarter the potatoes, and trim and halve the green beans.
Add the potatoes to the boiling water and cook about 5 minutes. You want the potatoes to be beginning to soften, but not be fully cooked. Add the green beans to the pot and cook for an additional 3-4 minutes, again you want the green beans to just begin to soften.
Add the pasta to the pot with the potatoes and green beans and cook until aldente. Reserve 1/2 cup of the pasta water.
In a large bowl toss the warm pasta mixture with the fresh pesto. Working a few tablespoons at a time, add the warm pasta water until the pesto has emulsified into a creamy sauce and has evenly coated the pasta.
Serve warm with additional grated cheese! Enjoy!