Bring a large pot of water to a boil, salt it, add the pasta and cook for 5 minutes. Add the potatoes and cook for 5 more minutes, then stir in the peas for 1 minute and drain, reserving 1 ladleful of the pasta cooking water.
While the pasta is working, using a food processor, combine the basil, tarragon, pistachios, vegetable broth, garlic and lemon peel; season with salt and pepper. Process, pouring in the EVOO in a steady drizzle, until coarsely pureed. Transfer to a large bowl and stir in the reserved pasta cooking water. Add the pasta, potatoes and peas and toss vigorously for 1 minute to coat the pasta. Top with the cheese and toss well.